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Math Help - Help with DE

  1. #1
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    Help with DE

    Please could someone help me with this diff equation - I have been trying multitudes of substitution and have not yielded any progress. Help will be very appreciated.

    x^2 y' - 2yx = \dfrac{y^2}{x^2} + \dfrac{2y}{x} + x^2 + 1

    Thank you friends
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  2. #2
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    Re: Help with DE

    That's a rather non-linear equation. There are NO general methods for solving such an equation. There are some methods for getting an approximate solution. Is that sufficient/
    Thanks from topsquark and Diadem
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  3. #3
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    Re: Help with DE

    Nevermind
    Last edited by Diadem; July 20th 2013 at 03:37 AM.
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  4. #4
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    Re: Help with DE

    Quote Originally Posted by Diadem View Post
    Please could someone help me with this diff equation - I have been trying multitudes of substitution and have not yielded any progress. Help will be very appreciated.

    x^2 y' - 2yx = \dfrac{y^2}{x^2} + \dfrac{2y}{x} + x^2 + 1

    Thank you friends
    Maybe a substitution of the form \displaystyle \begin{align*} v = \frac{y}{x} \end{align*} would work.

    Notice that if \displaystyle \begin{align*} v = \frac{y}{x} \end{align*}, then \displaystyle \begin{align*} y = v\,x \end{align*} and \displaystyle \begin{align*} \frac{dy}{dx} = v + x\,\frac{dv}{dx} \end{align*}, so substituting into the DE we have

    \displaystyle \begin{align*} x^2\,\frac{dy}{dx} - 2\,x\,y &= \frac{y^2}{x^2} + \frac{2\,y}{x} + x^2 + 1 \\ x^2 \left( v + x\,\frac{dv}{dx} \right) - 2\,x\left( v\,x \right) &= v^2 + 2v + x^2 + 1 \\ x^2\,v + x^3\,\frac{dv}{dx} - 2\,x^2 \, v &= v^2 + 2\,v + x^2 + 1 \\ x^3\,\frac{dv}{dx} - x^2\,v &= v^2 + 2\,v + x^2 + 1 \\ \frac{dv}{dx} - x^{-1}\,v &= x^{-3}\,v^2 + 2\,x^{-3}\,v + x^{-1} + x^{-3} \\ \frac{dv}{dx} + \left( -x^{-1} - 2\,x^{-3} \right) v &= x^{-3}\,v^2 + x^{-1} + x^{-3} \end{align*}

    This is ALMOST a Bernoulli DE, I wonder if anyone knows how to proceed from here...
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  5. #5
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    Re: Help with DE

    Quote Originally Posted by Diadem View Post
    Please could someone help me with this diff equation - I have been trying multitudes of substitution and have not yielded any progress. Help will be very appreciated.
    x^2 y' - 2yx = \dfrac{y^2}{x^2} + \dfrac{2y}{x} + x^2 + 1
    Thank you friends
    Hi !
    This is a Riccati equation.
    write it on the form : y' = P(x) +Q(x)*y +R(x)*y
    Let y = -z'/(z*R)
    Compute y' in terms of z'', z', z
    Bring back y' and y into y' = P(x) +Q(x)*y +R(x)*y
    This leads to a linear ODE (second order)
    Solve it for z(x)
    Compute z'
    Bring back z' and z into y = -z'/(z*R) which leads to y(x)
    Thanks from topsquark
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