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Integrating Factor for First Order Linear Differential Equation

  1. Oct 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Find an integrating factor for the first order linear differential equation
    [itex] \frac{dy}{dx} - \frac{y}{x} = xe^{2x} [/itex]
    and hence find its general solution

    2. Relevant equations



    3. The attempt at a solution
    I found the integrating factor which is [itex]e^{-lnx} = x^{-1}[/itex]

    and multiplying the equation with the integrating factor, will result in:
    [itex]\frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = e^{2x}[/itex]

    how do I go on from here?
     
  2. jcsd
  3. Oct 25, 2012 #2

    tiny-tim

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    hi wowmaths! :smile:

    the LHS is the exact derivative of … ? :wink:
     
  4. Oct 25, 2012 #3

    HallsofIvy

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    Do you know why you found the "integrating factor"?

    The whole point of an integrating factor for [itex]dy/dx+ a(x)y= f(x)[/itex] is that, with integrating factor [itex]\mu(x)[/itex], we will have
    [tex]\mu(x)\frac{dy}{dx}+ \mu(x)a(x)y= \frac{d(\mu(x)y}{dx}= \mu(x)f(x)[/tex]

    If [itex]\mu(x)= 1/x[/itex] here (I have not checked that) then your equation should reduce to
    [tex]\frac{d(y/x)}{dx}= e^{2x}[/tex]
    Integrate both sides of that with respect to x.
     
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