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Thread: ODE proof type question

  1. #1
    Junior Member
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    ODE proof type question

    Hi, I am struggling with the following question:

    if yp1 satisfies y''p1 + p(x)y'p1 + q(x)yp1 = r1(x), yp2 satisfies y''p2 + p(x)y'p2 + q(x)yp2 = r2(x) and yh satisfies y''h + p(x)y'h + q(x)yh = 0 , show that y = yh+ yp1 + yp2 satisfies y'' + p(x)y' + q(x)y = r1(x) + r2(x)

    Usually I would post my attempt at a question on here but I actually don't know how to get started on this one. I am doing okay actually using this principle to solve ODEs so far, but I don't know how to prove it.

    Thanks for any help, and I hope the above is readable.

    Andy
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  2. #2
    MHF Contributor
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    Re: ODE proof type question

    Hey andy000.

    Hint: Try taking your solution, getting its first and second derivatives and then plugging into the differential equation and showing that it holds.
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  3. #3
    Junior Member
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    Re: ODE proof type question

    Hi thanks for your reply,

    So does this mean just saying y = yh + yp1 + yp2, therefore y' = y'h + y'p1 + y'p2 , y'' = y''h + y''p1 + y''p2 and subbing in to the left side of the ODE?

    Then this would give :

    y''h + y''p1 + y''p2 + p(x)(y'h + y'p1 + y'p2) +q(x)(yh + yp1 + yp1) = 0 + r1(x) +r2(x)

    Is that all there is to this or am i missing something?
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