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About Solvable/Unsolvable ODEs

  1. Mar 4, 2014 #1
    In my class, I learned about a First-order ODEs,

    and solvable and unsolvable.

    example in case solvable ODEs)
    dy/dt=t/y
    dy/dt=y-t^2

    example in case unsolvable ODEs)
    dy/dt=t-y^2

    but , i dont know how distinguish those.

    plz, teach ME! : ( as possible as easily !
     
  2. jcsd
  3. Mar 4, 2014 #2

    hilbert2

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    Science Advisor
    Gold Member

    The question is not about whether the equation has a solution. It's about how easy it is to find the solution. The equation ##\frac{dy}{dt}=\frac{t}{y}## is separable and is easy to solve by multiplying both sides with ##ydt## and integrating the resulting equation ##ydy=tdt##. The equation ##\frac{dy}{dt}=y-t^{2}## can be solved by first multiplying both sides with the integrating factor ##e^{-t}##, and then using the derivative of product rule and integrating the resulting equation ##\frac{d}{dt}\left(e^{-t}y\right)=t^{2}e^{-t}##.

    For the nonlinear and non-separable equation ##\frac{dy}{dt}=t-y^{2}##, there is no similar simple method of solution. There does exist a solution, but it must be written in terms of special functions called Bessel functions. Do you know how to solve DE:s with WolframAlpha or Mathematica?
     
  4. Mar 4, 2014 #3
    Oh,! I got it! Thanks : )
     
    Last edited: Mar 4, 2014
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