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<h2 class="hd hd-2 unit-title">4.1. Complex cover-up</h2>
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Carry out this process to determine the coefficients [mathjaxinline]c,d[/mathjaxinline] of the second partial fraction. </p>
<table id="a0000000467" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\frac{20(2s+3)}{(s^2+25)(s^2+2s+3)}= \frac{as+b}{s^2+25}+\frac{c(s+1)+d}{(s+1)^2+2}\, .[/mathjax]</td>
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<p style="display:inline">[mathjaxinline]c=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]d=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">4.2. Solve a differential equation.</h2>
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Activity.
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<h4 onclick="hideshow(this);" style="margin: 0px">Laplace Table<span class="icon-caret-down toggleimage"/></h4>
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<table id="a0000000475" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
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&#160;
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[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
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<td style="width:20%; border:none;text-align:right" class="eqnnum">(3.75)</td>
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[mathjaxinline]\displaystyle e^{rt}[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
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&#160;
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[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
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<td style="width:20%; border:none;text-align:right" class="eqnnum">(3.76)</td>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \cos \omega t[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
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&#160;
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[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
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<td style="width:20%; border:none;text-align:right" class="eqnnum">(3.77)</td>
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[mathjaxinline]\displaystyle \sin \omega t[/mathjaxinline]
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[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
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&#160;
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[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(3.78)</td>
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[mathjaxinline]\displaystyle t[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
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&#160;
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[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(3.79)</td>
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Solve the initial value problem [mathjaxinline]\dot x+3x=2\cos (4t)[/mathjaxinline] with initial condition [mathjaxinline]x(0)=1[/mathjaxinline], using Laplace transform. </p>
<p>
First find a formula for [mathjaxinline]X[/mathjaxinline] in terms of [mathjaxinline]s[/mathjaxinline]: <p style="display:inline">[mathjaxinline]X=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_rec4-tab2-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Determine [mathjaxinline]x[/mathjaxinline] by using partial fractions and the inverse Laplace transform. Use the Laplace table of values above. </p>
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<p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p>
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<tbody>
<tr class="fiptitle">
<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" scope="row" rowspan="3">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>
</td>
</tr>
<tr class="formulainput">
<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" scope="row" rowspan="4">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" scope="row" rowspan="3">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>
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<h2 class="hd hd-2 unit-title">4.3. Practice with rest initial conditions.</h2>
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Activity.
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<p>
A spring/mass/dashpot system driven through the spring is modeled by the equation </p>
<table id="a0000000501" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]m\ddot x + b \dot x + kx = ky \, .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<ul class="itemize">
<li>
<p>
Case 1: [mathjaxinline]x(0) = 0[/mathjaxinline], &#8201; &#8201; [mathjaxinline]\dot x(0) = 0.[/mathjaxinline] </p>
<p>
Find the Laplace transform [mathjaxinline]X(s)[/mathjaxinline], the Laplace transform of [mathjaxinline]x(t)[/mathjaxinline] in terms of [mathjaxinline]Y[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], and the system parameters. </p>
<p>
(Type <b class="bf">Y</b> for [mathjaxinline]Y(s)[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]X(s)=[/mathjaxinline]</p>
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\(\)
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<p>
Case 2: For comparison let's look at a case that does not have rest initial conditions. <br/>[mathjaxinline]x(0) = 0[/mathjaxinline], &#8201; &#8201; [mathjaxinline]\dot x(0) = 1.[/mathjaxinline] </p>
<p>
Find the Laplace transform [mathjaxinline]X(s)[/mathjaxinline], the Laplace transform of [mathjaxinline]x(t)[/mathjaxinline] in terms of [mathjaxinline]Y[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], and the system parameters. </p>
<p>
(Type <b class="bf">Y</b> for [mathjaxinline]Y(s)[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]X(s)=[/mathjaxinline]</p>
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\(\)
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<tr class="fiptitle">
<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" scope="row" rowspan="3">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>
</td>
</tr>
<tr class="formulainput">
<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" scope="row" rowspan="4">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" scope="row" rowspan="3">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>
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