Find a solution with Laplacetransform

**Muffin** · October 2nd 2011, 02:53 AM

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

With given data the equation becomes

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

Solving Y(s)

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

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.com 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!!

**CaptainBlack** · October 2nd 2011, 03:20 AM

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

With given data the equation becomes

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

Solving Y(s)

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

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.com 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!!

Your LT of the translated unit step is wrong

$\mathcal{L}[u(t-\tau)]=\frac{e^{-\tau s}}{s}$

CB

**Muffin** · October 2nd 2011, 03:52 AM

Okey thx! But still.. Something is wrong. I dont get the right answer

**CaptainBlack** · October 2nd 2011, 03:55 AM

Originally Posted by Muffin

Okey thx! But still.. Something is wrong. I dont get the right answer

So what do you now get?

CB

**Muffin** · October 2nd 2011, 04:02 AM

I get the same. When I inverstransform $\frac{81e^{\frac{-\pi }{2}}}{s}$ I get $81U(t-\frac{\pi }{2})sin(9t)$

**CaptainBlack** · October 2nd 2011, 06:26 AM

Originally Posted by Muffin

I get the same. When I inverstransform $\frac{81e^{\frac{-\pi }{2}}}{s}$ I get $81U(t-\frac{\pi }{2})sin(9t)$

What is:

$\mathcal{L}^{-1}\left[ \frac{1}{s^2+a^2}\right]$ ?

That may not get you all the way to the solution but it might help.

CB

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