From the class notes:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\begin{array}{l}

y'' + 8y' + 16y = te^{ - 4t} ,\,\,\,\,\,y\left( 0 \right) = y'\left( 0 \right) = 0 \\

\\

L\left[ {y''} \right] + 8L\left[ {y'} \right] + 16L\left[ y \right] = \frac{1}{{\left( {s + 4} \right)^2 }} \\

\end{array}[/tex]

How did he get [tex]\frac{1}{{\left( {s + 4} \right)^2 }}[/tex] ?

From the table, [tex]t = \frac{1}{{s^2 }}[/tex] and [tex]e^{at} \to \frac{1}{{s - a}}[/tex]

How do these combine to give [tex]\frac{1}{{\left( {s + 4} \right)^2 }}[/tex] ?

The next line is

[tex]s^2 y\left( s \right) - sy\left( 0 \right) - y'\left( 0 \right) + 8\left( {sy\left( s \right) - y\left( 0 \right) + 16y\left( s \right)} \right) = \frac{1}{{\left( {s + 4} \right)^2 }}[/tex]

Where did everything on the left side of = come from? The table doesn’t have y’’ or y’.

After this, the problem looks like it turns into algebra.

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# Homework Help: Laplace Transform

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