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Homework Help: Differential Equations homework problem.

  1. May 27, 2010 #1
    1. The problem statement, all variables and given/known data
    The equation [tex]\frac{dy}{dx}[/tex] = A(x)y2+B(x)y+C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known. Show that the substitution

    y = y1+[tex]\frac{1}{v}[/tex]

    transforms the Riccati equation into the linear equation

    [tex]\frac{dv}{dx}[/tex]+ (B+2Ay1)v = -A.


    3. The attempt at a solution

    So i know that y' = y1' - [tex]\frac{v'}{v^2}[/tex] and I have tried substituting y and y' back into the original equation in order to simplify it down to the linear equation given. Unfortunately for some reason I am not getting anywhere. Any help would be appreciated!

    Thanks,
    Rob

    P.S. Sorry for the crappy formatting, not really sure what I am doing.
     
  2. jcsd
  3. May 27, 2010 #2

    Mark44

    Staff: Mentor

    When you make your substitution, the left side will be dy1/dx - v'/v. Are you remembering to replace dy1/dx by A(x)y12 + B(x)y1 + C(x)?
     
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