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Math Help - Differential Equation...transformation?

  1. #1
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    Post Differential Equation...transformation?

    the reason for the ? in the title is because I'm not really sure what method to use here:

    Determine v(x) such that the transformation y=u(x)v(x) will transform y''+p(x)y'+q(x)y=0 into u''+g(x)u=0 where g(x) can be expressed in terms of p(x) and q(x). Your answer should express both v(x0 and g(x) in terms of p(x) and q(x).

    I have tried the following:

    Let y=uv (callign the functions u and v for simplicity's sake)
    Then y'=uv' + vu'
    y'' = uv''+2u'v'=u''v

    Then I plugged these back into the original DE, and after grouping got:
    u''v+u'(2v'+pv)+u(v''+pv'+qv)=0.

    I have no idea where to go from here though. any thoughts are greatly appreciated. Thanks so much!
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  2. #2
    Moo
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    Hello,

    Okay, you're up to this point :
    u''v+u'(2v'+pv)+u(v''+pv'+qv)=0.
    Now divide by v (if v=0, it's a nonsense) :

    u''+u' \cdot \frac{2v'+pv}{v}+u(\frac{v''}{v}+p \frac{v'}{v}+q)=0

    So since in the final expression, there is no more u', solve for v in 2v'+pv=0

    then substitute it in \frac{v''}{v}+p \frac{v'}{v}+q and it'll give you g =)
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  3. #3
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    Ah! Thank you very much! Rest of the process was easy enough...you gave me the missing pieces. Thanks again!
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