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Solve the differential equation

  1. Nov 22, 2009 #1
    1. The problem statement, all variables and given/known data
    a.Find solution to the differential equation
    dy/dx=cos(x^2)*exp(sin(x));y(0)=0 for x in the interval [0,10]
    b.find y(10)


    2. Relevant equations



    3. The attempt at a solution
    I don't know where to begin
     
  2. jcsd
  3. Nov 22, 2009 #2
    That differential equation is separable.
     
  4. Nov 22, 2009 #3
    I got:
    y = cos(x^2)*exp(sin(x))
    integral(y) = integral(cos(x^2)*exp(sin(x)))
    I got stuck. What do I need to do next
     
  5. Nov 22, 2009 #4

    Mark44

    Staff: Mentor

    After separation you should have
    [tex]\int dy~=~\int cos(x^2)e^{sin(x)}dx[/tex]

    Now is a good time to verify that you have given us the correct differential equation.
     
  6. Nov 22, 2009 #5
    I actually use ODE45 in matlab to solve the equation and plot it simultaneously.
    I tried to integrate the equation using 'int' command but it did not work.
    I have no clue how to solve it with only one variable on the right hand side because to solve the separable differential equation you need x and y.
     
  7. Nov 22, 2009 #6
    are u sure this is the right differential equation?

    u get [tex]Y =~\int cos(x^2)e^{sin(x)}dx[/tex]

    but i enter the right side in mathematica and get no result
     
  8. Nov 22, 2009 #7
    I am positive. The original equation is dy/dx = cos(x^2)*exp(sin(x))
     
  9. Nov 22, 2009 #8

    Mark44

    Staff: Mentor

    For the a part,
    [tex]y(x)~=~\int_{t = 0}^{x} cos(t^2)e^{sin(t)}dt[/tex]

    For the b part,
    [tex]\int_{x = 0}^{10} dy~=~\int_{x = 0}^{10} cos(x^2)e^{sin(x)}dx[/tex]
    [tex]\Rightarrow y(10) - y(0)~=~\int_{x = 0}^{10} cos(x^2)e^{sin(x)}dx[/tex]
    Since y(0) = 0, then
    [tex]y(10)~=~\int_{x = 0}^{10} cos(x^2)e^{sin(x)}dx[/tex]

    I don't think you can do much more with this if the exact solution is what is wanted.
     
  10. Nov 22, 2009 #9
    But how can I find y(10)
     
  11. Nov 22, 2009 #10

    Mark44

    Staff: Mentor

    Look in post #9. There it is.
     
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