Solve the differential equation.
-$$\frac { d y } { d x } - \frac { y } { x } = ( \ln x ) ^ { 5 }$$
A) $y = \frac { 1 } { 6 } ( \ln x ) ^ { 6 } + C x$
B) $y = x ( \ln x ) ^ { 6 } + C x$
C) $y = \frac { 1 } { 6 } x ( \ln x ) ^ { 6 } + C x$
D) $y = \frac { 1 } { 6 } x ^ { 6 } + C x$

Question

Quizplus · Accepted Answer

This differential equation is a first-order linear differential equation which can be solved using an integrating factor. The integrating factor is $e^{\int -\frac{1}{x}dx} = e^{-\ln|x|} = \frac{1}{x}$. Multiplying both sides of the differential equation by this integrating factor transforms it into an exact differential, which can then be integrated directly. The solution involves integrating $\frac{1}{x}(\ln x)^5$ with respect to $x$, which gives $\frac{1}{6}(\ln x)^6$. Multiplying by the integrating factor's inverse (which is $x$) and adding the general solution of the homogeneous equation ($C x$) gives the final answer: $y = \frac{1}{6}x(\ln x)^6 + Cx$.

Solve the Differential Equation dydx−yx=(ln⁡x)5\frac { d y } { d x } - \frac { y } { x } = ( \ln x ) ^ { 5 }dxdy​−xy​=(lnx)5

Solve the Differential Equation $\frac { d y } { d x } - \frac { y } { x } = ( \ln x ) ^ { 5 }$