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Math Help - Solving 1st order DE

  1. #1
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    Solving 1st order DE

    How would you go about solving this DE: dy/dx=2xy/(x^2-3y^2) .....could you turn it into an exact differential....ie 2xy dx- (x^2-3y^2) dy=0.....I don't think this will work.
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    Re: Solving 1st order DE

    Perhaps a substitution of the form \displaystyle \begin{align*} v = \frac{y}{x} \implies y = v\,x \implies \frac{dy}{dx} = v + x\,\frac{dv}{dx} \end{align*}, giving

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

    Since the variables have been separated, you can now solve using integration.
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    Re: Solving 1st order DE

    Thanks for that
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    Re: Solving 1st order DE

    Quote Originally Posted by heatly View Post
    How would you go about solving this DE: dy/dx=2xy/(x^2-3y^2) .....could you turn it into an exact differential....ie 2xy dx- (x^2-3y^2) dy=0.....I don't think this will work.
    The reason is, of course, that this NOT an "exact differential". The derivative of 2xy, with respect to y, is 2x while the derivative of -(x^2- 3y^2), with respect to x, is -2x.

    Every first order equation, however, has an "integrating factor". Here, that integrating factor is \frac{1}{y^2}. Multiplying that equation by \frac{1}{y^2} gives 2\frac{x}{y}dx- \left(\frac{x^2}{y^2}- 3\right)dy= 0. (Notice that this is now clearly in terms of \frac{x}{y} so Prove It's substitution works.)

    The derivative of \frac{2x}{y}, with respect to y, is -\frac{2x}{y^2} and the derivative of -\left(\frac{x^2}{y^2}- 3\right), with respect to x, is also -\frac{2x}{y^2}.

    That means that there exist a function, F(x,y), such that dF= \frac{\partial F}{\partial x}dx+ \frac{\partial F}{\partial y}dy= \frac{2x}{y}dx- \left(\frac{x^2}{y^2}- 3\right)dy= 0.

    Since \frac{\partial F}{\partial x}= \frac{2x}{y}, F(x, y)= \frac{x^2}{y}+ g(y) for some function, g, of y only. Differentiating that with respect to y, \frac{\partial F}{\partial y}= -\frac{x^2}{y^2}+ g'= -\frac{x^2}{y^2}+ 3 so that g(y)= 3y. That is, F(x,y)= \frac{x^2}{y}+ 3y and since "dF= 0" means that F is a constant, the solution to the differential equation is \frac{x^2}{y}+ 3y= C.
    Last edited by HallsofIvy; August 8th 2013 at 06:08 PM.
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    Re: Solving 1st order DE

    Thanks a lot
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