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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Differential equations</h3>
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<h2 class="hd hd-2 unit-title">2. Differential equations</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Recognize that first order <span style="color:#27408C"><b class="bf">differential equations</b></span> describe a relationship between a function and its derivative, and that solutions are found using anti-differentiation. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">separation of variables</b></span> to find <span style="color:#27408C"><b class="bf">families of solution curves</b></span>. </p></li><li><p>
Solve differential equations with <span style="color:#27408C"><b class="bf">initial conditions</b></span>. </p></li><li><p>
Draw and utilize <span style="color:#27408C"><b class="bf">slope fields</b></span> of differential equations to gain qualitative understanding of solutions to differential equations. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">Euler's Method</b></span> to approximate solutions to differential equations. </p></li></ul><p><h3>Contents: 13 pages</h3></p><p>
9 videos (40 minutes 1x speed) 20 questions 2 mathlets </p>
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<h2 class="hd hd-2 unit-title">3. Introducing differential equations</h2>
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<h3 class="hd hd-2">Introducing differential equations</h3>
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Units of the antiderivative
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<p>
Differential equations, like all equations, must have units that agree on both sides of the equal sign. </p>
<p>
Consider the differential equation [mathjaxinline]\displaystyle \frac{dx}{dt} = f(t)[/mathjaxinline], where [mathjaxinline]x[/mathjaxinline] has units of meters, and [mathjaxinline]t[/mathjaxinline] has units of seconds. </p>
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What are the units of [mathjaxinline]\displaystyle \int f(t) dt[/mathjaxinline]? </td>
<td style="text-align:left; border:none">
What are the units of [mathjaxinline]dt[/mathjaxinline]? </td>
<td style="text-align:left; border:none">
What are the units of [mathjaxinline]\displaystyle f(t) dt[/mathjaxinline]?</td>
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<input type="radio" name="input_antider_3-tab3-problem1_2_1" id="input_antider_3-tab3-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="antider_3-tab3-problem1_2_1-choice_1-label" for="input_antider_3-tab3-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_2_1">
<text> meters</text>
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<input type="radio" name="input_antider_3-tab3-problem1_2_1" id="input_antider_3-tab3-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="antider_3-tab3-problem1_2_1-choice_2-label" for="input_antider_3-tab3-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_2_1">
<text> seconds</text>
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<input type="radio" name="input_antider_3-tab3-problem1_2_1" id="input_antider_3-tab3-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab3-problem1_2_1-choice_3-label" for="input_antider_3-tab3-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_2_1">
<text> meters per second</text>
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<input type="radio" name="input_antider_3-tab3-problem1_2_1" id="input_antider_3-tab3-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="antider_3-tab3-problem1_2_1-choice_4-label" for="input_antider_3-tab3-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_2_1">
<text> meter[mathjaxinline]\cdot[/mathjaxinline]seconds</text>
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<input type="radio" name="input_antider_3-tab3-problem1_2_1" id="input_antider_3-tab3-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="antider_3-tab3-problem1_2_1-choice_5-label" for="input_antider_3-tab3-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_2_1">
<text> unitless</text>
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<span id="answer_antider_3-tab3-problem1_2_1"/>
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<span class="status unanswered" id="status_antider_3-tab3-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_antider_3-tab3-problem1_3_1">
<fieldset aria-describedby="status_antider_3-tab3-problem1_3_1">
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<input type="radio" name="input_antider_3-tab3-problem1_3_1" id="input_antider_3-tab3-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="antider_3-tab3-problem1_3_1-choice_1-label" for="input_antider_3-tab3-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_3_1">
<text> meters</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_antider_3-tab3-problem1_3_1" id="input_antider_3-tab3-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="antider_3-tab3-problem1_3_1-choice_2-label" for="input_antider_3-tab3-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_3_1">
<text> seconds</text>
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<input type="radio" name="input_antider_3-tab3-problem1_3_1" id="input_antider_3-tab3-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab3-problem1_3_1-choice_3-label" for="input_antider_3-tab3-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_3_1">
<text> meters per second</text>
</label>
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<div class="field">
<input type="radio" name="input_antider_3-tab3-problem1_3_1" id="input_antider_3-tab3-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="antider_3-tab3-problem1_3_1-choice_4-label" for="input_antider_3-tab3-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_3_1">
<text> meter[mathjaxinline]\cdot[/mathjaxinline]seconds</text>
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<input type="radio" name="input_antider_3-tab3-problem1_3_1" id="input_antider_3-tab3-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="antider_3-tab3-problem1_3_1-choice_5-label" for="input_antider_3-tab3-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_3_1">
<text> unitless</text>
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<span id="answer_antider_3-tab3-problem1_3_1"/>
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<span class="status unanswered" id="status_antider_3-tab3-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<input type="radio" name="input_antider_3-tab3-problem1_4_1" id="input_antider_3-tab3-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="antider_3-tab3-problem1_4_1-choice_1-label" for="input_antider_3-tab3-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_4_1">
<text> meters</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_antider_3-tab3-problem1_4_1" id="input_antider_3-tab3-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="antider_3-tab3-problem1_4_1-choice_2-label" for="input_antider_3-tab3-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_4_1">
<text> seconds</text>
</label>
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<div class="field">
<input type="radio" name="input_antider_3-tab3-problem1_4_1" id="input_antider_3-tab3-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab3-problem1_4_1-choice_3-label" for="input_antider_3-tab3-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab3-problem1_4_1">
<text> meters per second</text>
</label>
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<text> meter[mathjaxinline]\cdot[/mathjaxinline]seconds</text>
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Units of the differential equation
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Consider the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)[/mathjaxinline], where [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]f(x)[/mathjaxinline] have units of meters ([mathjaxinline]m[/mathjaxinline]). </p>
<p>
What are the units of the solution [mathjaxinline]\displaystyle y = \int f(x) dx[/mathjaxinline]? </p>
<p>
(Enter [mathjaxinline]m[/mathjaxinline] for meters, [mathjaxinline]s[/mathjaxinline] for seconds, and none if the solution has no units.) </p>
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<font color="#0078b0">2520</font>
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Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">5. Separation of variables</h2>
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<p>
In the example we just saw, we were given a differential equation of the form [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)g(y)[/mathjaxinline]. To solve, we separated all terms involving [mathjaxinline]x[/mathjaxinline] from terms involving [mathjaxinline]y[/mathjaxinline], and applied anti-differentiation to solve. </p><table id="a0000000521" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000522"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{dy}{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = f(x) g(y)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000523"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{dy}{g(y)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = f(x) dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000524"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \frac{dy}{g(y)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int f(x) dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table>
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Separable differential equations
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Which of the following differential equations are separable? </p>
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(Choose all that can be rewritten in the form [mathjaxinline]y' = f(x)g(y)[/mathjaxinline].) </p>
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<text>[mathjaxinline]\displaystyle \frac{dy}{dx} + x = xy[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle y' =e^ y \cdot \cos x[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle y' - xy^2 = 0[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{dy}{dx} = \cos (xy)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle x(y^{\prime })^2 - y = 0[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle y' = x+y[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">6. Separation of variables practice</h2>
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Separation of variables practice 1
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Solve the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = xy^2[/mathjaxinline]. </p>
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(Enter C for the constant of anti-differentiation.) </p>
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<p style="display:inline">[mathjaxinline]y=[/mathjaxinline]</p>
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Separation of variables practice 2
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Solve the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = (2x+5)^4[/mathjaxinline]. </p>
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(Enter C for the constant of anti-differentiation.) </p>
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<p style="display:inline">[mathjaxinline]y=[/mathjaxinline]</p>
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<font color="#0078b0">2520</font>
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Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-2">Recitation video: separation of variables</h3>
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Checking units
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Consider the differential equation [mathjaxinline]\ \displaystyle \frac{dy}{dx} = y[/mathjaxinline]. Suppose that [mathjaxinline]y[/mathjaxinline] has units of meters. What are the units of [mathjaxinline]x[/mathjaxinline]? </p>
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<text> meters per second</text>
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Solving the differential equation
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Use separation of variables to solve the differential equation [mathjaxinline]\displaystyle \ \frac{dy}{dx} = y[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">8. Initial conditions</h2>
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<p><h3>Theorem</h3> Given a differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)g(y)[/mathjaxinline] and an <span style="color:#99182C"><b class="bf">initial condition</b></span> [mathjaxinline]y(a) = b[/mathjaxinline], if [mathjaxinline]\ f[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and [mathjaxinline]g'[/mathjaxinline] are continuous near [mathjaxinline](a,b)[/mathjaxinline], then there is a unique function [mathjaxinline]y[/mathjaxinline] whose derivative is given by [mathjaxinline]f(x)g(y)[/mathjaxinline] and that passes through the point [mathjaxinline](a,b)[/mathjaxinline]. </p>
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Initial condition practice 1
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Solve the differential equation with the given initial condition. </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{dy}{dx} = 4xy, \qquad y(0) = 10[/mathjax]</td>
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<div class="formulainput">
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<tbody>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Initial condition practice 2
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{dy}{dx} = \sqrt {y+1}, \qquad y(0) = 3[/mathjax]</td>
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<div class="formulainput">
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Geometric example practice 1
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Find the curves perpendicular to the parabolic curves from the example above. </p>
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This statement is giving us a statement about the slopes of curves. Which differential equation best describes this statement of slope? </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{dy}{dx} =[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
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<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Geometric example practice 2
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Find the general solution to the differential equation you found in the problem above. </p>
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<text> [mathjaxinline]x^2 + y^2 = C[/mathjaxinline]</text>
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<text> [mathjaxinline]2x^2 - y^2 = C[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{x^2}{2} + y^2 = C[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{x^2}{4} - y = C[/mathjaxinline]</text>
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The parameter [mathjaxinline]C[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_antider_3-tab9-problem2_3_1">
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<input type="radio" name="input_antider_3-tab9-problem2_3_1" id="input_antider_3-tab9-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="antider_3-tab9-problem2_3_1-choice_1-label" for="input_antider_3-tab9-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab9-problem2_3_1"> <text> [mathjaxinline]&gt; 0[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab9-problem2_3_1" id="input_antider_3-tab9-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="antider_3-tab9-problem2_3_1-choice_2-label" for="input_antider_3-tab9-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab9-problem2_3_1"> <text> [mathjaxinline]\geq 0[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab9-problem2_3_1" id="input_antider_3-tab9-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab9-problem2_3_1-choice_3-label" for="input_antider_3-tab9-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab9-problem2_3_1"> <text> [mathjaxinline]&lt; 0[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab9-problem2_3_1" id="input_antider_3-tab9-problem2_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="antider_3-tab9-problem2_3_1-choice_5-label" for="input_antider_3-tab9-problem2_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab9-problem2_3_1"> <text> [mathjaxinline]\neq 0[/mathjaxinline]</text>
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Geometric example practice 3
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When there is a unique solution, solution curves are: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_antider_3-tab9-problem3_2_1">
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Which of the following initial conditions will not determine a unique solution curve? (Check all that apply.) </p>
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<h2 class="hd hd-2 unit-title">10. Slope fields</h2>
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Consider the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y.[/mathjaxinline] Without solving the differential equation, what is the slope of a solution curve through the point [mathjaxinline](0, 1)[/mathjaxinline]? Hint: Use the differential equation and the geometric interpretation of the derivative? </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \left. \frac{dy}{dx}\right|_{(0,1)}=[/mathjaxinline]</p>
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<p><b class="bfseries">Slope fields</b></p><p><span style="color:#99182C">Note on video:</span> In the video, it says that solutions become parallel to the [mathjaxinline]x[/mathjaxinline]-axis. It should really say that solutions become asymptotic to the [mathjaxinline]x[/mathjaxinline]-axis. </p><p>
The <span style="color:#99182C"><b class="bf">slope field</b></span> is a diagram that helps us to visualize the information in a first order differential equation. The slope field is obtained as follows. At each point [mathjaxinline](x,y)[/mathjaxinline], you draw a short segment whose slope is the <em>value</em> of [mathjaxinline]\displaystyle y'[/mathjaxinline] at the point [mathjaxinline](x,y)[/mathjaxinline]. The solution curves must be tangent to the slope field at all points. </p>
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Identify the slope field
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Which of the following is the slope field for [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y[/mathjaxinline]? </p>
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<img alt="Slope field in the x y plane. Points along the x axis have zero slope. Points below the x axis have positive slope. Points above the x axis have positive slope." src="/assets/courseware/v1/fc835e33383f23729e5894a917acb664/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot4.svg" style="margin: 10px 25px 25px 25px" width="375px"/>
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<img alt="Slope field in the x y plane. Points along the line y equals negative x have zero slope. Points above this line have positive slope. Points below this line have negative slope." src="/assets/courseware/v1/9fdf89b30d608af23972ebd7ad954271/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot2.svg" style="margin: 10px 25px 25px 25px" width="375px"/>
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<img alt="Slope field in the x y plane. Points along the lines y equals x and y equals negative x have zero slope. Points above y equals x and below y equals negative x have positive slope. Points above y equals negative x and below y equals x have positive slope. Points above both line have negative slope. Points below both line have negative slope." src="/assets/courseware/v1/1916b8f6a2dbd497bfb552405713dac0/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot3.svg" style="margin: 10px 25px 25px 25px" width="375px"/>
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<img alt="Slope field in the x y plane. Points along the x and y axes have zero slope. Points in the first and third quadrants have positive slope. Points in the second and fourth quadrants have negative slope." src="/assets/courseware/v1/87202337af75a114e69211c5b66c7041/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot6.svg" style="margin: 10px 25px 25px 25px" width="375px"/>
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<h3 class="hd hd-3 problem-header" id="antider_3-tab10-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@antider_3-tab10-problem3-problem-progress" tabindex="-1">
Identify the solution curves 1
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<iframe height="630 px" src="https://mathlets1801.surge.sh/slopeFields.html" style="border:0px; display: block;" width="1100 px"/>
<p>
Use the dropdown menu in the mathlet above to find the slope field for the differential equation [mathjaxinline]\displaystyle y' = y(1-y)[/mathjaxinline]. </p>
<p>
Click on the point given in the initial condition to see the solution curve through that point. What happens to the given solution curve as [mathjaxinline]x[/mathjaxinline] tends towards infinity? </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
[mathjaxinline]y(0)=1/2[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]y(0) = -1[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]y(0) = 2[/mathjaxinline]</td>
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<text> Solution approaches [mathjaxinline]\infty[/mathjaxinline]</text>
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<text> Solution approaches [mathjaxinline]1[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_antider_3-tab10-problem3_2_1" id="input_antider_3-tab10-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab10-problem3_2_1-choice_3-label" for="input_antider_3-tab10-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_2_1">
<text> Solution approaches [mathjaxinline]0[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab10-problem3_2_1" id="input_antider_3-tab10-problem3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="antider_3-tab10-problem3_2_1-choice_4-label" for="input_antider_3-tab10-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_2_1">
<text> Solution approaches [mathjaxinline]-\infty[/mathjaxinline]</text>
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<text> Solution approaches [mathjaxinline]\infty[/mathjaxinline]</text>
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<text> Solution approaches [mathjaxinline]1[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab10-problem3_3_1" id="input_antider_3-tab10-problem3_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab10-problem3_3_1-choice_3-label" for="input_antider_3-tab10-problem3_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_3_1">
<text> Solution approaches [mathjaxinline]0[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab10-problem3_3_1" id="input_antider_3-tab10-problem3_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="antider_3-tab10-problem3_3_1-choice_4-label" for="input_antider_3-tab10-problem3_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_3_1">
<text> Solution approaches [mathjaxinline]-\infty[/mathjaxinline]</text>
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<text> Solution approaches [mathjaxinline]\infty[/mathjaxinline]</text>
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<text> Solution approaches [mathjaxinline]1[/mathjaxinline]</text>
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<input type="radio" name="input_antider_3-tab10-problem3_4_1" id="input_antider_3-tab10-problem3_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="antider_3-tab10-problem3_4_1-choice_3-label" for="input_antider_3-tab10-problem3_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_4_1">
<text> Solution approaches [mathjaxinline]0[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_antider_3-tab10-problem3_4_1" id="input_antider_3-tab10-problem3_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="antider_3-tab10-problem3_4_1-choice_4-label" for="input_antider_3-tab10-problem3_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem3_4_1">
<text> Solution approaches [mathjaxinline]-\infty[/mathjaxinline]</text>
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<h3 class="hd hd-3 problem-header" id="antider_3-tab10-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@antider_3-tab10-problem4-problem-progress" tabindex="-1">
Identify the solution curve 2
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<iframe height="630 px" src="https://mathlets1801.surge.sh/slopeFields.html" style="border:0px; display: block;" width="1100 px"/>
<p>
Use the dropdown menu in the mathlet above to find the slope field for the differential equation [mathjaxinline]\displaystyle y' = y/x[/mathjaxinline]. </p>
<p>
Click on the point [mathjaxinline](0,1)[/mathjaxinline] to see the solution curve through that point. Which choice best describes the solution curve? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_antider_3-tab10-problem4_2_1">
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<input type="radio" name="input_antider_3-tab10-problem4_2_1" id="input_antider_3-tab10-problem4_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="antider_3-tab10-problem4_2_1-choice_1-label" for="input_antider_3-tab10-problem4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_antider_3-tab10-problem4_2_1">
<text> A ray coming out of the origin along the positive [mathjaxinline]y[/mathjaxinline]-axis.</text>
</label>
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Application of linear approximation
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<p>
Consider the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y[/mathjaxinline]. Find the linear approximation for the solution function [mathjaxinline]\ f(x)[/mathjaxinline] that passes through the point [mathjaxinline](0, 1)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]f(x)\approx[/mathjaxinline]</p>
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<p>
Use the approximation above to estimate a value of [mathjaxinline]\ f(0.1).[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]f(0.1)\approx[/mathjaxinline]</p>
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What is the best approximation for the solution to the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y[/mathjaxinline] through the point [mathjaxinline](0.1,f(0.1))[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]f(x)\approx[/mathjaxinline]</p>
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<p>
Use the approximation above to estimate a value of [mathjaxinline]\ f(0.2).[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]f(0.2)\approx[/mathjaxinline]</p>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<p>
Congratulations! You just carried out the first two steps of Euler's Method. We can repeat this process to come up with an iterative formula for the value of the function on successive intervals. </p><p>
Given the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y[/mathjaxinline]: </p><ol class="enumerate"><li value="1"><p>
Choose a step size [mathjaxinline]h[/mathjaxinline] (The smaller the step size, the more accurate the approximation.) </p></li><li value="2"><p>
Let [mathjaxinline](x_0, y_0)[/mathjaxinline] be the initial condition. </p></li><li value="3"><p>
We can use the differential equation and step size to determine the value of the function at [mathjaxinline]x_1=x+h[/mathjaxinline]: </p><table id="a0000000580" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000581"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x_{1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = x_0 + h[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000582"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y_{1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = y_0 + (x_0 + y_0)h \quad \textrm{linear approximation}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table></li><li value="4"><p>
Iterate this process: </p><table id="a0000000583" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000584"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x_{k+1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = x_ k + h[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000585"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y_{k+1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = y_ k + (x_ k + y_ k)h \quad \textrm{linear approximation}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table></li></ol><p>
Once we have an iterative formula, this is something that we can program into a computer! The following python script is the process a computer uses to apply Euler's method for as many steps, and with as small a step size as you want. </p><pre>
#set initial condition
x,y = x0, y0
#set step size
h=0.1
#count iterations of method
stepcounter = 0
#determine how many steps to take
while stepcounter < 1000:
y= y+(x+y)*h
x+=h
stepcounter+= 1
</pre><p>
The smaller the step [mathjaxinline]h[/mathjaxinline], the better the approximation. You can visualize Euler's method by using the mathlet below. Use the dropdown menu to choose a differential equation, choose a step size, and click on the graph to select an initial value. Watch as the process unfolds! See what happens for different values of [mathjaxinline]h[/mathjaxinline]. </p><iframe src="https://mathlets1801.surge.sh/eulersMethod.html" width="1100 px" height="630 px" style="border:0px; display: block;"/>
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<h2 class="hd hd-2 unit-title">12. More practice solving differential equations</h2>
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The solution to a differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)g(y)[/mathjaxinline] is a <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_antider_3-tab12-problem1_2_1">
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Identify the initial conditions which do not specify a unique solution function to the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = \frac{xy}{x^2-1}[/mathjaxinline]. </p>
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Method of educated guessing
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Using the method of educated guessing, which of the following functions solve the differential equation [mathjaxinline]\displaystyle \frac{d^2y}{dx^2} = -y[/mathjaxinline]? </p>
<p>
(Choose all functions that satisfy the differential equation.) </p>
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<text>[mathjaxinline]Ae^{-x}[/mathjaxinline]</text>
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<text>[mathjaxinline]B\ln (x)[/mathjaxinline]</text>
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<text>[mathjaxinline]C\sin (x)[/mathjaxinline]</text>
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<text>[mathjaxinline]D\cos (x)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">13. Summary</h2>
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<p><b class="bfseries">Separation of variables</b></p><p>
A differential equation is <span style="color:#99182C"><b class="bf">separable</b></span> if it can be written in the form [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)g(y)[/mathjaxinline]. </p><p>
To solve a separable differential equation, separate all terms involving [mathjaxinline]x[/mathjaxinline] from terms involving [mathjaxinline]y[/mathjaxinline], and anti-differentiate both sides. </p><table id="a0000000586" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000587"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{dy}{dx}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = f(x) g(y)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000588"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{dy}{g(y)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = f(x) dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000589"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \frac{dy}{g(y)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int f(x) dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p><b class="bfseries">Initial conditions</b></p><p><b class="bfseries">Theorem</b></p><p>
Given a differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = f(x)g(y)[/mathjaxinline] and an <span style="color:#99182C"><b class="bf">initial condition</b></span> [mathjaxinline]y(a) = b[/mathjaxinline], if [mathjaxinline]\ f[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and [mathjaxinline]g'[/mathjaxinline] are continuous near [mathjaxinline](a,b)[/mathjaxinline], then there is a unique function [mathjaxinline]y[/mathjaxinline] whose derivative is given by [mathjaxinline]f(x)g(y)[/mathjaxinline] and that passes through the point [mathjaxinline](a,b)[/mathjaxinline]. </p><p><b class="bfseries">Slope fields</b></p><p>
The <span style="color:#99182C"><b class="bf">slope field</b></span> is a diagram that helps us to visualize the information in a first order differential equation. The slope field is obtained as follows. At each point [mathjaxinline](x,y)[/mathjaxinline], you draw a short segment whose slope is the <em>value</em> of [mathjaxinline]\displaystyle y'[/mathjaxinline] at the point [mathjaxinline](x,y)[/mathjaxinline]. The solution curves must be tangent to the slope field at all points. </p><p><b class="bfseries">Euler's method</b></p><p>
Given the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = x+y[/mathjaxinline]: </p><ol class="enumerate"><li value="1"><p>
Choose a step size [mathjaxinline]h[/mathjaxinline] (The smaller the step size, the more accurate the approximation.) </p></li><li value="2"><p>
Let [mathjaxinline](x_0, y_0)[/mathjaxinline] be the initial condition. </p></li><li value="3"><p>
We can use the differential equation and step size to determine the value of the function at [mathjaxinline]x_1=x+h[/mathjaxinline]: </p><table id="a0000000590" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000591"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x_{1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x_0 + h[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000592"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y_{1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y_0 + (x_0 + y_0)h \quad \textrm{linear approximation}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table></li><li value="4"><p>
Iterate this process: </p><table id="a0000000593" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000594"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x_{k+1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x_ k + h[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000595"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y_{k+1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y_ k + (x_ k + y_ k)h \quad \textrm{linear approximation}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table></li></ol>
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