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Math Help - How to solve this ODE?..

  1. #1
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    How to solve this ODE?..

    I'm reading a book on asymptotic methods and I encountered the following ODE, do you have any clue how to solve this? Thanks.

    y''+y=\frac{\cos(x)}{5+8\cos(x)\sin(x)}
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  2. #2
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    Re: How to solve this ODE?..

    Hey Mengqi.

    You should try using the complem entary and particular solution.

    On the right hand side you have a 2nd order linear DE which can be solved through a standard technique. On the Right hand side you have a particular solution which can be solved using a number of techniques including operator methods (using the d/dx = D as an operator).

    These techniques should be taught in any standard differential equations courses.
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  3. #3
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    Re: How to solve this ODE?..

    Thanks Chiro,

    I see. But when I tried to solve it in this way: I know the left hand side will give me the solution like y=A\cos(x)+B\sin(x) (with right hand side being set to zero temporally), then I assume y_{particular}=C(x)(A\cos(x)+B\sin(x)) and plug it in, but this way seems not to work. In the end, I got sth. like [C''(x)(A\cos x+B\sin x)-2C'(x)(A\cos x+B\sin x )](5+8\cos x\sin x)=\cos x. How to proceed from here? I didn't reduce the problem to a simple one, C still appears with a second order derivative. Do you know any other techniques? Thanks
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  4. #4
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    Re: How to solve this ODE?..

    And it seems not trivial because I just used Mathematica to see the final result, which is attached. I think it involves some advanced technique? It's great if any of you could solve it. I am curious to see the derivation. Thanks.
    Attached Thumbnails Attached Thumbnails How to solve this ODE?..-capture-d-ecran-2014-01-19-10.36.26.png  
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