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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:
    v=x-y
    y=x-v
    y'=1-dv/dx

    1-dv/dx=sin(v)
    1-sin(v)=dv/dx
    dx=dv/(1-sin(v))
    x=2/(cot(v/2)-1)

    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.
    Thanks!
     
  2. jcsd
  3. Feb 4, 2009 #2
    f =

    1/(1-sin(x))


    >> int(f,x)

    ans =

    -2/(tan(1/2*x)-1)

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

    Mark44

    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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