# Homework Help: Find a solution with Laplace

1. Oct 2, 2011

### Muffin

1. The problem statement, all variables and given/known data
Determine the solution for
$$y^{''}+81y=81U(t-\frac{\pi }{2})$$
when $$\left\{y(0)=12,y'(0)=18\right\}$$

U(t) is the unit step function

3. The attempt at a solution

Laplacetransforming :
$$s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=$$$$\frac{81e^{\frac{-\pi }{2}}}{s}$$

With given data the equation becomes

$$s^{2}Y(s)-12s-18+81Y(s)=$$$$\frac{81e^{\frac{-\pi }{2}}}{s}$$

Solving Y(s)

$$Y(s)=81e^{-\frac{\pi }{2}s}(\frac{1}{s(s^{2}+9^{2})})+12(\frac{s}{{s^{2}+9^{2}}})+18(\frac{1}{s^{2}+9^{2}})$$

Transform again:
$$y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t$$

Problem: This is wrong but I dont know what am I doing wrong..Can someone tell me what Im doing wrong? Wolframalpha says that the solution should be: $$U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1$$

But I dont understand how to get that answer.

Thanks

Last edited: Oct 2, 2011