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Math Help - Bernoulli Differential Equation

  1. #1
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    Bernoulli Differential Equation

    Find the general solution to the differential equation. Primes denote derivatives with respect to x throughout.

    (2xsin(y)cos(y))y' = 4x^2 + (sin(y))^2
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    Make the substitution u=\sin^2y so that u'=2\sin(y)\cos(y)y' and the ODE is u'x=4x^2+u which is linear.
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    I was able to get this far but I am still having trouble. Should the integrating factor be -(1/x)
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  4. #4
    Air
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    Quote Originally Posted by bearej50 View Post
    I was able to get this far but I am still having trouble. Should the integrating factor be -(1/x)
    It's close. You have a sign error.

    u'x=4x^2+u
    u'x - u =4x^2
    u' - \frac{u}{x} =4x

    IF = e^{-\int \frac{1}{x} \ \mathrm{d}x} = e^{-\ln x} = e^{\ln |x^{-1}|} = x^{-1} = \frac{1}{x}
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  5. #5
    MHF Contributor matheagle's Avatar
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    Any multiple of an integrating factor suffices.
    That's why you don't need the +C when you integrate.
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