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Math Help - Changing equation into ODE... easy?

  1. #1
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    Unhappy Changing equation into ODE... easy?

    Find an ODE of the form y''+ay'+by=0 where a^2>4b for which the given function is a solution, or else explain why no such ODE exists.

    The first ones I have been able to figure out by reverse-engineering the solving process...

    e.g. e^t+exp(2t) \longrightarrow y''-3y'+2y=0

    However, I am having trouble with these two:
    • t^2 e^t
    • exp(x^2)


    Any insight would be greatly appreciated!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by horan View Post
    Find an ODE of the form y''+ay'+by=0 where a^2>4b for which the given function is a solution, or else explain why no such ODE exists.

    The first ones I have been able to figure out by reverse-engineering the solving process...

    e.g. e^t+exp(2t) \longrightarrow y''-3y'+2y=0

    However, I am having trouble with these two:
    • t^2 e^t
    • exp(x^2)

    Any insight would be greatly appreciated!
    say for t^2e^t

    let y = t^2 e^t

    find y' and y'' then plug them into the form of the equation you were given. then solve for a,b such that a^2 > 4b
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  3. #3
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    Thanks for the reply!. Unfortunately, this yields a really messy slew of algebra :-(

    I get:
    4 t e^t+e^t(2+t^2)+a t e^t (t+2)+b t^2 e^t = 0

    ... but I cannot figure out how to prove that a^2>4b.
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