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? asked in Science & MathematicsMathematics · 2 months ago

Consider the differential equation xv dv dx + v 2 = 32x 2 . (a) Solve the differential equation using an appropriate integrating factor.?

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  • 2 months ago

    Your typography is weird.  Did you mean

    xv dv/dx + v^2 = 32x^2 ?

    First I'll solve it using v = xu, dv/dx = u + x du/dx.

    ux^2 (u + x du/dx) + x^2u^2 = 32x^2,

    so wherever x is not 0, we get

    u(u + x du/dx) + u^2 = 32 =>

    u^2 + xu du/dx + u^2 = 32 =>

    xu du/dx = 32 - u^2 =>

    u du/(32 - u^2) = dx/x =>

    -(1/2)ln(32 - u^2) = ln(x) + C1 =>

    1/sqrt(32 - u^2) = cx =>

    1/sqrt(32 - v^2/x^2) = cx, or

    x*sqrt(32 - v^2/x^2) = C.

    Can I go back and solve the equation with an integrating factor?  It's not coming to me right now...

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