Integrating Factor.

**krzyrice** · May 26th 2010, 11:57 PM

cos x dy/dx + y sin x = sin x cos x

dy/dx - y/x = (ln x)^4

How would i solve for find y in these two problems? I'm looking over a similar example which used substitution but I'm confused on these two.

**Prove It** · May 27th 2010, 01:22 AM

Originally Posted by krzyrice

cos x dy/dx + y sin x = sin x cos x

dy/dx - y/x = (ln x)^4

How would i solve for find y in these two problems? I'm looking over a similar example which used substitution but I'm confused on these two.

$\cos{x}\,\frac{dy}{dx} + y\sin{x} = \sin{x}\cos{x}$

$\frac{dy}{dx} + y\tan{x} = \sin{x}$

This is now first order linear, so the integrating factor is

$e^{\int{\tan{x}\,dx}} = e^{\ln{\sec{x}}}= \sec{x}$ .

Multiplying through by the integrating factor gives:

$\sec{x}\,\frac{dy}{dx} + y\,\tan{x}\sec{x} = \tan{x}$

$\frac{d}{dx}(y\,\sec{x}) = \tan{x}$

$y\,\sec{x} = \int{\tan{x}\,dx}$

$y\,\sec{x} = \ln{|\sec{x}|} + C$

$y = \cos{x}\ln{|\sec{x}|} + C\cos{x}$ .

**Prove It** · May 27th 2010, 01:28 AM

Originally Posted by krzyrice

cos x dy/dx + y sin x = sin x cos x

dy/dx - y/x = (ln x)^4

How would i solve for find y in these two problems? I'm looking over a similar example which used substitution but I'm confused on these two.

$\frac{dy}{dx} - \frac{y}{x} = (\ln{x})^4$ .

This is first order linear, so the integrating factor is

$e^{-\frac{1}{x}\,dx} = e^{-\ln{x}} = e^{\ln{(x)^{-1}}} = x^{-1}$ .

Multiplying through by the integrating factor gives:

$x^{-1}\,\frac{dy}{dx} - x^{-2}y = x^{-1}(\ln{x})^4$

$\frac{d}{dx}(x^{-1}y) = x^{-1}(\ln{x})^4$

$x^{-1}y = \int{(\ln{x})^4\,x^{-1}\,dx}$ .

Now make the substitution $u = \ln{x}$ so that $du = x^{-1}\,dx$ , the equation becomes

$x^{-1}y = \int{u^4\,du}$

$x^{-1}y = \frac{u^5}{5} + C$

$x^{-1}y = \frac{(\ln{x})^5}{5} + C$

$y = \frac{x(\ln{x})^5}{5} + Cx$ .

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