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Solving a differential equation

  1. May 2, 2014 #1
    1. The problem statement, all variables and given/known data
    Solve
    [tex](1+bx)y''(x)-ay(x)=0[/tex]


    2. Relevant equations



    3. The attempt at a solution

    I'm used to solving homogeneous linear ODE's where you form a characteristic equation and solve that way, here there is the function of x (1+bx) so how does that change things?
     
  2. jcsd
  3. May 2, 2014 #2
    Would dividing both sides by 1+bx help?
     
  4. May 2, 2014 #3
    Ok so if I did that then what? I can define a characteristic equation such that

    [tex]r^2-\frac{a}{1+bx}=0[/tex]

    and [tex]r=\pm\sqrt{\frac{a}{1+bx}}[/tex]

    where [tex] b^2-4ac = 4a(1+bx) > 0[/tex]

    so a solution is [tex]y=ce^{rx}[/tex] but that doesn't satisfy the ODE so its not correct?
     
  5. May 2, 2014 #4

    LCKurtz

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    You can't use constant coefficient methods on a DE like this with variable coefficients. Perhaps there is a clever substitution that will help, or maybe not. Problems like this are typically solved with series solutions, especially if you know ##a## and ##b##. Where did this equation come from? If it's from a text, the recent material may give a hint how to solve it.
     
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