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First order nonlinear ODE - Integrating factor + exact differentials, or not?

  1. Jun 24, 2010 #1
    First order nonlinear ODE -- Integrating factor + exact differentials, or not?

    Hello everyone,

    (I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question).

    I recently came across the following nonlinear ODE that I am having difficulty in solving.

    1. The problem statement, all variables and given/known data

    Determine the general solution to the differential equation:

    [tex]\frac{dy}{dx} = \frac{x^2 - y^2}{x^2 + 5yx} = \frac{(x+y)(x-y)}{x(x + 5y)}[/tex]

    2. Relevant equations

    Any methods up to and including a typical advanced engineering mathematics (undergrad) course (integrating factors, variation of parameters, transforms, exact differentials, etc.)

    3. The attempt at a solution

    My only ideas have involved forcing the equation to be an exact differential. However, I have not been able to come to a solution by this route. To demonstrate, I pursued putting the equation in the form of conventional notation for exact differentials,

    [tex]
    \underbrace{(x^2 - y^2)}_{=\, M}dx - \underbrace{(x^2 + 5yx)}_{=\, -N}dy = 0
    [/tex]

    [tex]
    Mdx + Ndy = 0
    [/tex]

    where

    [tex]M = x^2 - y^2 \Rightarrow M_x = 2x, \quad M_y = -2y[/tex]
    [tex]N = -(x^2 + 5yx) \Rightarrow N_x = -2x - 5x \quad N_y = 5x[/tex]

    and the subscripts denote differentiation with respect to the labeled parameter. It is evident it is not exact ([tex]M_y \neq N_x[/tex]). Further, I was unable to implement an integrating factor [tex]\sigma[/tex] such that [tex]\frac{\partial}{\partial y}(\sigma M) = \frac{\partial}{\partial x} (\sigma N)[/tex] upon multiplication of the entire equation by [tex]\sigma[/tex]. It can readily be seen that the integrating factor [tex]\sigma[/tex] is not only a function of [tex]x[/tex] or [tex]y[/tex], e.g. writing out the condition for exact differentials involving the factor [tex]\sigma[/tex], and enforcing [tex]\sigma[/tex] to be a function of [tex]x[/tex] or [tex]y[/tex] alone involves results of the form

    [tex]\frac{\sigma_y}{\sigma} \sim \frac{M_y - N_x}{M}[/tex]

    or

    [tex]\frac{\sigma_x}{\sigma} \sim \frac{M_y - N_x}{N}[/tex]

    which demonstrates that [tex]\sigma[/tex] is a function of both variables in either case (in contradiction to the posits that lead to these equations).

    Choosing a form [tex]\sigma (x,y) = x^a y^b[/tex], I could not discern proper constants [tex]a[/tex], and [tex]b[/tex].

    Does anyone have any suggestions regarding solving this equation? Thanks very much for any insight, and take care.
     
  2. jcsd
  3. Jun 25, 2010 #2

    Mark44

    Staff: Mentor

    Re: First order nonlinear ODE -- Integrating factor + exact differentials, or not?

    No need for apologies - what you have looks great.
    Try this substitution: Let v = y/x ==> y = vx ==> y' = v'x + v

    Then the DE y' = (x^2 - y^2)/(x^2 + 5xy) becomes
    v'x + v = (x^2 - v^2 x)/(x^2 + 5vx^2)
    [tex]v'x + v = \frac{x^2 - v^2x^2}{x^2 + 5vx^2}[/tex]

    After a bit of algebraic manipulation, you get a DE in v and x that is separable. Undo the substitution and you have your solution.
     
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