Ok we are given the ODE(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

{y}^{\prime\prime}(t) + \omega^2{y(t)} = {r(t)}

[/tex]

[tex]

r(t) = cos\omega{t}

[/tex]

[tex] \omega = 0.5,0.8,1.1,1.5,5.0,10.0

[/tex]

I know you can use variation of paramaters to solve for it so I start by finding the complementary solution.

[tex]

{y}^{\prime\prime}(t) + \omega^2{y(t)} = 0

[/tex]

We know solutions are of the form

[tex]

y = \exp{(mt)}

[/tex]

so after taking derivatives and what not we get the fundamental solution

[tex]

\cos\omega{t}, \sin\omega{t}

[/tex]

Our complementary solution is

[tex]

{y}_{c}=Acos \omega{t} + Bsin \omega{t}

[/tex]

For the particular solution we set

[tex]

{y}^{\prime\prime}(t) + \omega^2{y(t)} = cos\omega{t}

[/tex]

We then use a fourier series to expand

[tex]

cos\omega{t}

[/tex]

Then proceed to solve for it but the problem I'm having is that I'm getting the fourier series to be zero which is strange. I know that there will be no

[tex]

{b}_{n}

[/tex]

term since cos is even but its still werid why I'm getting zero for

[tex]

{a}_{0}, {a}_{n}

[/tex]

Any help would be appreciated.

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# ODE with Fourier Series

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