Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$
A) $y = \frac { x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$
B) $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 11 } }$
C) $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$
D) $y = \frac { 11 x ^ { 2 } } { 4 } + \frac { C } { x ^ { 9 } }$

Question

Quizplus · Accepted Answer

The given differential equation is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = \frac{9}{x}$ and $Q(x) = 4x$. To solve it, we first find the integrating factor (IF), which is $e^{\int P(x)dx} = e^{\int \frac{9}{x}dx} = e^{9\ln|x|} = |x|^9$. Multiplying both sides of the differential equation by the integrating factor, we integrate to find the general solution. The correct solution involves integrating $4x|x|^9$, leading to the term $\frac{4x^2}{11}$ as part of the general solution, and the integrating factor $|x|^9$ appears in the denominator of the arbitrary constant term, making the correct choice $y = \frac{4x^2}{11} + \frac{C}{x^9}$.

Find the General Solution of the Given First-Order Linear Differential dydx+9yx=4x\frac { d y } { d x } + \frac { 9 y } { x } = 4 xdxdy​+x9y​=4x

Find the General Solution of the Given First-Order Linear Differential $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$