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Is there a general way of solving ODE's of the form f(y,y')=0?

  1. Nov 16, 2011 #1
    Is there a general proposed way of solving ODE's of the form f(y,y')=0? any ideas?
     
  2. jcsd
  3. Nov 17, 2011 #2
    Solve the equation f(a,t)=0 for t, considering "a" as a parameter. The result is one ore several functions t=g(a)
    Let t=y' and a=y
    y'=g(y)
    For each function g :
    dy/dx = g(y)
    integrate (1/g(y))dy = dx
    the result is on the forme x = f(y)
    calculate x = reciprocal function of f(y)

    Example :
    y²+y'²-1 = 0
    equation to be solved : f(a,t) = a²+t²-1 = 0
    t = sqrt(1-a²)
    dy/dx = sqrt(1-y²)
    dx = dy/sqrt(1-y²)
    x = arcsin(y) +C
    y = sin(x-C)

    However, some difficulties might be encountered :
    - If analytical solving of equation f(a,t)=0 is not possible.
    - if a primitive of the function 1/g(x) is not known
    - if the reciprocal of function x=f(y) cannot be analytically computed.
     
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