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Math Help - first order homogenous equations

  1. #1
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    first order homogenous equations

    We've just started differential equations and the subsitution we use is y=vx, I was wondering whether thats the only substitution to use or whether there are others...and if there are others, how will I know which one to use?

    Thanks
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  2. #2
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    Re: first order homogenous equations

    Either y=ux, with dy=u\,dx+x\,du, or x=vy, with dx=v\,dy+y\,dv, will reduce a first-order homogeneous equation to a separable one. Here's a quote from Zill, 6th Ed., p. 55:

    Although either of the indicated substitutions can be used for every homogeneous differential equation, in practice we try x=vy whenever the function M(x,y) is simpler than N(x,y).
    Note that in Zill's notation, the DE

    M(x,y)\,dx+N(x,y)\,dy=0

    is the homogeneous DE under discussion.

    Does that answer your question?
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  3. #3
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    Re: first order homogenous equations

    Yes there are others. The only way to know which to use is experience.
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    Re: first order homogenous equations

    Ahh thank you...it makes a bit more sense now! So what would I use for (3x^2)y(dy/dx)= x^3 + 2y^3

    Thanks again
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    Re: first order homogenous equations

    Quote Originally Posted by maths13 View Post
    Ahh thank you...it makes a bit more sense now! So what would I use for (3x^2)y(dy/dx)= x^3 + 2y^3

    Thanks again
    Well, what does the DE look like in the form of Post # 2?
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  6. #6
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    Re: first order homogenous equations

    Quite frankly, it is probably simplest just to always use y= xv rather than take the time to worry about it!
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