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General tips on making useful substitutions for differential equations? watch

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    Hi all, my maths isn't brilliant but usually if i'm given a substitution, I can figure out a solution fairly easily, for a ordinary differential equation

    But I'm very bad at figuring what a useful substitution to make is. Let me give a few examples, along with the substitution suggested in the answers, and can someone explain why that substituion is the logical choice? Thanks alot!

     \dfrac{dy}{dx}-y=xy^5
    suggested substitution:  u= y^{-4}

     \dfrac{dy}{dx}+y=y^2 (cos(x)-sin(x))
    suggested substitution:  u= y^{-1}

    Can't see the logic :confused:
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    I think both these equations are of the form described here
    http://en.wikipedia.org/wiki/Bernoul...ntial_equation

    Basically if you have something of the form  \frac{dy}{dx} + P(x)y = Q(x)y^n then substitute  u = y^{-(n-1)} .

    There are probably other specific types of ODEs which have a specific useful substitution, but I can't remember them if I covered them in my course. We did cover these Bernoulli ones though.
 
 
 
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Updated: January 28, 2010
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