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Math Help - Use substitution to solve a differential equation

  1. #1
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    Use substitution to solve a differential equation

    By using the substitution v=x-y, solve the differential equation \frac{dy}{dx}+[1+(x-y)^2]cos^2x=sin^2x, expressing y in terms of x.

    \frac{dv}{dx}=1-\frac{dy}{dx}

    1-\frac{dv}{dx}+[1+v^2]cos^2x=sin^2x

    How do I continue to remove the x in the trigo?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Use substitution to solve a differential equation

    Quote Originally Posted by Punch View Post
    By using the substitution v=x-y, solve the differential equation \frac{dy}{dx}+[1+(x-y)^2]cos^2x=sin^2x, expressing y in terms of x.

    \frac{dv}{dx}=1-\frac{dy}{dx}

    1-\frac{dv}{dx}+[1+v^2]cos^2x=sin^2x

    How do I continue to remove the x in the trigo?
    \frac{dv}{dx}=1+\cos^2x+v^2\cos^2x-\sin^2x

    =2\cos^2x+v^2\cos^2x

    =\cos^2x(v^2+2)

    \frac{dv}{v^2+2}=\cos^2xdx

    Can you proceed?
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  3. #3
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    Re: Use substitution to solve a differential equation

    Quote Originally Posted by alexmahone View Post
    \frac{dv}{dx}=1+\cos^2x+v^2\cos^2x-\sin^2x

    =2\cos^2x+v^2\cos^2x

    =\cos^2x(v^2+2)


    \frac{dv}{v^2+2}=\cos^2xdx

    Can you proceed?
    Yeap, thanks!
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