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First order non-linear non-homogeneous DE problem

  1. May 2, 2012 #1
    Hello. I am trying to solve this problem methodically but my solution does not seem to agree with the given answer.

    1. The problem statement, all variables and given/known data

    The differential equation is:
    (sinx)y' - (cosx)y = 1 + C

    2. Relevant equations



    3. The attempt at a solution

    y' - (cosx/sinx)y = 1/sinx + C/sinx

    When finding the integrating factor, I used:
    e^-∫(cosx/sinx)dx
    I was wondering if this was either
    |sinx|^(-1) or -|sinx| and why? I tried solving the equation using both methods

    1st method (using |sinx|^(-1))

    (|sinx|^(-1))y' - cosx/((sinx)^2).y = 1/(sinx)^2 + C/(sinx)^2
    LHS becomes
    D[|sinx|^(-1).y] = 1/(sinx)^2 + C/(sinx)^2
    Integrating both sides and multiplying by |sinx| gives
    y = x/sinx + Cx/sinx + Dsinx

    When using -|sinx| I get:
    y = x/sinx Cx/sinx + D/sinx

    The given solution is
    y = -x.cosx + sinxln(|sinx|) - Acosx + Bsinx

    I have no idea how to get this. Any help would be appreciated, thanks!

    Edit: I see my error for using the |sinx|^(-1), I didn't integrate the right hand side properly.

    I get |sinx|^(-1).y = -cotx + Ccotx + D
    so
    y = -xcosx + Ccosx + Dsinx, which is still not the right answer
     
    Last edited: May 2, 2012
  2. jcsd
  3. May 2, 2012 #2
    It is in fact linear and on the form:
    [itex] y' + f(x)y+g(x) = 0[/itex]
    The solution should be as follows:
    [itex] y(x) = \left(\int g(x)e^{\int f(x)\text{d}x}\text{d}x+K_1\right)e^{\int (-f(x))\text{d}x}[/itex]
     
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