Bernoulli Differential Equation

**bearej50** · February 20th 2009, 06:57 AM

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

**Krizalid** · February 20th 2009, 07:20 AM

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.

**bearej50** · February 20th 2009, 09:35 AM

I was able to get this far but I am still having trouble. Should the integrating factor be -(1/x)

**Air** · February 20th 2009, 11:02 AM

Originally Posted by bearej50

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.

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

$IF = e^{-\int \frac{1}{x} \ \mathrm{d}x} = e^{-\ln x} = e^{\ln |x^{-1}|} = x^{-1} = \frac{1}{x}$

**matheagle** · February 22nd 2009, 09:25 PM

Any multiple of an integrating factor suffices.
That's why you don't need the +C when you integrate.

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