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Solving problem analytically

  1. Aug 11, 2014 #1
    1. The problem statement, all variables and given/known data
    Differential equation
    ## \frac{df}{dx}=\sin f(x)+\cos x ##



    2. Relevant equations



    3. The attempt at a solution
    If I integrate equation I will get
    ## f(x)=\int \sin f(x)dx+\sin x+C ##
    is there any possibility to solve that analytically?
     
  2. jcsd
  3. Aug 11, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    Almost certainly not. There is no general solution method for the ODE [itex]dy/dx = F(x,y)[/itex] subject to [itex]y(x_0) = y_0[/itex] except in special cases, which amount to when [itex]dy/dx = p(x)q(y)[/itex] or there is a change of dependent and/or independent variable which will put the ODE into that form. You then have
    [tex]
    \int_{y_0}^{y(x)} \frac1{q(s)}\,ds = \int_{x_0}^{x} p(t)\,dt
    [/tex]
    and you then have to ask whether you can (a) do those integrals analytically, and (b) solve the resulting equation for [itex]y(x)[/itex] analytically.

    If the answer to either of those questions is "no", then it may be easier to solve the ODE numerically using a suitable method, and if you can't turn your ODE into a separable equation then numerical methods are your only recourse.
     
  4. Aug 11, 2014 #3
    Is it possible that there's a typo in the problem statement, and that it should be sin(x) f(x) on the rhs, rather than sin f(x) ?

    Chet
     
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