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  <question type="description">
    <name>
      <text>20.3.2.2 Example Laplace with Derivatives</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>If \(\displaystyle{{\cal L}\{f(t)\}=F(s)\quad \text{ and}\quad {d^2f\over {dt^2}}-{df\over {dt}}=3t}\) with initial conditions \(f(0)=1,\quad f'(0)=0\), find the explicit expression for \(F(s)\).</p><br>
<ol class="HELM_parts">
<li>
<p>Begin by finding \(\displaystyle{{\cal L}\biggl\{{d^2f\over {dt^2}}\biggr\},\ \ {\cal L}\biggl\{{df\over {dt}}\biggr\}\  \text{ and}\ {\cal L}\{3t\}}\)</p>
<h4 class="HELM_solution">Solution</h4>
<p>\[\begin{eqnarray*}
{\cal L}\{3t\}&amp;=&amp;3/s^2\\
{\cal L}\biggl\{{df\over
{dt}}\biggr\}&amp;=&amp;-f(0)+sF(s)=-1+sF(s)\\
{\cal L}\biggl\{{d^2f\over
{dt^2}}\biggr\}&amp;=&amp;-f'(0)-sf(0)+s^2F(s)=-s+s^2F(s)
\end{eqnarray*}\]</p>
</li>
<li>
<p>Now complete the calculation to find \(F(s)\):</p>
<h4 class="HELM_solution">Solution</h4>
<p>You should find \(\quad\displaystyle {F(s)={s^3-s^2+3\over s^3(s-1)}}\)<br />
since, using the transforms we have found: \[\begin{eqnarray*}
-s+s^2F(s)-(-1+sF(s))&amp;=&amp;{3\over {s^2}}\\\text{ so}\qquad
F(s)[s^2-s]&amp;=&amp;{3\over {s^2}}+s-1={s^3-s^2+3\over {s^2}}
\end{eqnarray*}\] leading to \( \displaystyle F(s)={s^3-s^2+3\over s^3(s-1)}\)</p>
</li>
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