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Homework Help: DE I don't get

  1. Oct 25, 2005 #1
    1. Use substitution x = e^t to transfer equation into one with constant coefficients and solve:
    x^2 y'' -3xy' + 13y = 4 + 3x
    My work:
    Okay, we have
    e^2t y'' - 3e^t y' + 13y = 4 + 3e^t.

    Now I am absolutely stuck. No idea how to solve it. Help!
  2. jcsd
  3. Oct 25, 2005 #2


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    Science Advisor

    You didn't complete the substitution! Your y' and y" are still with respect to x, not t.

    Use the chain rule: dy/dx= dy/dt dt/dx. Since x= et, t= ln x and
    dt/dx= 1/x. That is, dy/dx= (1/x)(dy/dt).
    d2y/dx2= d((1/x)(dy/dt)/dx= (-1/x2)dy/dt+ (1/x)d(dy/dt)/dx= ((-1/x2)dy/dt+(1/x)(1/x)d2y/dt2=.
    Now, x2 y'' = d2y/dt2- dy/dt
    and -3xy'= -3 dy/dt
    so the differential equation is
    d2y/dt2- 4dy/dt+ 13y= 4+ 3ett

    That should be easy to solve!
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