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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Laplace Transform

**Physics Forums | Science Articles, Homework Help, Discussion**