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Homework Help: Substitution in Differential Equations

  1. Feb 4, 2009 #1
    Make an appropriate substitution to find a solution of the equation dy/dx=sin(x-y). Does this general solution contain the linear solution y(x)=x-pi/2 that is readily verified by substitution in the differential equation?

    Here's what I did:


    The solution in the back of the book gives:
    x=tan(x-y) + sec(x-y)

    What am I doing wrong?
    Any help is greatly appreciated.
  2. jcsd
  3. Feb 4, 2009 #2
    f =


    >> int(f,x)

    ans =


    I have something different from matlab when I integrate 1/(1-sin(x))
  4. Feb 5, 2009 #3


    Staff: Mentor

    Everything is fine to here, but goes downhill after that.
    The integral on the right isn't too bad.

    [tex]\int \frac{dv}{1 - sin(v)}[/tex]
    [tex]= \int \frac{dv}{1 - sin(v)} * \frac{1 + sin(v)}{1 + sin(v)}[/tex]
    [tex]=\int \frac{(1 + sin(v))dv}{1 - sin^2(v)}[/tex]

    The denominator simplifies to cos^2(v) and you can split the integral into two integrals, one of which is straightforward. The other one requires only an ordinary substitution.

    I ended with the same answer as in the book.

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