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Second Order Linear Differential Equation (Non-constant coefficients)

  1. Apr 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the general solution for the following differential equation:
    y'' + x(y')^2 = 0.


    2. Relevant equations
    Integration, differentiation...


    3. The attempt at a solution
    Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify such an equation to a first order DE. The (y')^2 part is throwing me off as this would give the equation:
    v(x)' + x(v(x))^2 = 0.
    This would be grand if it weren't for the v(x)^2 term. Any ideas on how to get around this? Thanks.
     
  2. jcsd
  3. Apr 27, 2013 #2

    CAF123

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    Gold Member

    Are you worried about the v^2 term because it makes the eqn non-linear? If so, there are ways to solve such an equation.
     
  4. Apr 27, 2013 #3
    Yeah. We've only covered linear DE's.
     
  5. Apr 27, 2013 #4

    CAF123

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    Gold Member

    Are you at all familiar with separation of variables?
     
  6. Apr 27, 2013 #5
    Yeah. So I'll have v'(x) = (-x)(v(x))^2 and I can then separate the variables and solve. Thanks!
     
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