# Math Help - Inverse Laplace using partial fractions

1. ## Inverse Laplace using partial fractions

So the question is an initial value problem involving a second order dif eq and we need to then use inverse Laplace.

y''+4y= cost y(0)=1 and y'(0)=0

expanding,

$s^2Y(s) - sy(0) -y'(0) + 4Y(s) = \frac{s}{s^2 +1}$

Then solve for Y(s)

$Y(s) = \frac{s}{(s^2 +4)(s^2+1)} - \frac{s}{s^2+4}$

the term on the right gives us cos2t

for the term on the left I used Partial fractions. our professor doesn't want us to solve for A,B,C etc. he just wants the functions. So following his example in class I did the following:

$\frac{As+B}{(s^2+1)} + \frac{Cs+D}{s^2+4}
\Rightarrow \frac{As}{s^2+1} + \frac{B}{s^2+1} + \frac{Cs}{s^2+4}+\frac{D}{s^2+4}
$

Using the table I got
$Y(s) = Acost + Bsint + (C-1)cos2t + Dsin2t$

The answer in the back of the book is
$\frac{1}{3}cos(2t) + \frac{2}{3}cos(t)$

Can anyone tell me what I'm doing wrong? I basically did EXACTLY what he did in class.

Thanks so much!

2. shouldn't Y(s) be +s/(s^2+4) rather than minus. You add s to the other side not subtract it.

3. Yes I'm sorry that should be a plus!

4. What values did you find for A, B, C, and D?