(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine the solution for

[tex]y^{''}+81y=81U(t-\frac{\pi }{2})[/tex]

when [tex]\left\{y(0)=12,y'(0)=18\right\}[/tex]

U(t) is the unit step function

3. The attempt at a solution

Laplacetransforming :

[tex]s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=[/tex][tex]\frac{81e^{\frac{-\pi }{2}}}{s}[/tex]

With given data the equation becomes

[tex]s^{2}Y(s)-12s-18+81Y(s)=[/tex][tex]\frac{81e^{\frac{-\pi }{2}}}{s}[/tex]

Solving Y(s)

[tex]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}})[/tex]

Transform again:

[tex]y(t)=81U(t-\frac{\pi }{2})sin(9t)+12cos9t+18sin9t[/tex]

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: [tex]U(\frac{\pi }{2}-t)(sin(9t)-1)+sin(9t)+12cos(9t)+1[/tex]

But I dont understand how to get that answer.

Thanks

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# Find a solution with Laplace

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