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Math Help - differential equation problem

  1. #1
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    differential equation problem

    Hi

    I have attached my question

    For part a) i got final answer as y=-4/5 +ce^(5/3x)

    i got this using intergrating factors

    However i cannot seem to do part b)


    could someone please help

    thank you
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  2. #2
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    x*y'+(1-2x)y=4x so y'+\frac{1-2x}{x}y=4

    The integrating factor is then e^{\int\frac{1-2x}{x}dx}

    So now, \int\frac{1-2x}{x}dx=\int\left(\frac{1}{x}-\frac{2x}{x}\right)dx=ln|x|-2x and we ignore the +c...

    So the integrating factor is e^{ln|x|-2x}=e^{ln|x|}e^{-2x}=xe^{-2x}

    So xe^{-2x}y'+e^{-2x}(1-2x)y=4xe^{-2x}

    and the left side of the equation is just \frac{dy}{dx}(xe^{-2x}y)

    So \frac{dy}{dx}(xe^{-2x}y)=4xe^{-2x}

    Now integrate both sides, but this time be sure to include the +c...

    xe^{-2x}y=-(2x+1)e^{-2x}+c

    and now you just have to use your initial condition to find c (that integration was done by parts which I trust you can do)
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  3. #3
    MHF Contributor Calculus26's Avatar
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