The general solution of $y ^ { 2 } d x - \left( x y + x ^ { 2 } \right) d y = 0$ is

A) $y e ^ { \frac { y } { x } } = c x$
B) $y e ^ { \frac { y } { x } } = c$
C) $y e ^ { \frac { y } { x } } = c x ^ { 2 }$
D) $y e ^ { \frac { y } { x } } = c x ^ { - 1 }$
E) $y e ^ { \frac { y } { x } } = c x ^ { - 2 }$

Question

The general solution of $y ^ { 2 } d x - \left( x y + x ^ { 2 } \right) d y = 0$ is

A) $y e ^ { \frac { y } { x } } = c x$
B) $y e ^ { \frac { y } { x } } = c$
C) $y e ^ { \frac { y } { x } } = c x ^ { 2 }$
D) $y e ^ { \frac { y } { x } } = c x ^ { - 1 }$
E) $y e ^ { \frac { y } { x } } = c x ^ { - 2 }$

Quizplus · Accepted Answer

This differential equation can be solved by separating variables or recognizing it as an exact differential equation. Rearranging gives $dx - \left( \frac{xy + x^2}{y^2} \right) dy = 0$. Integrating both sides with respect to an appropriate variable and simplifying should lead to the solution $y e^{\frac{y}{x}} = c$, where $c$ is the constant of integration. This form matches choice B.

The General Solution Of y2dx−(xy+x2)dy=0y ^ { 2 } d x - \left( x y + x ^ { 2 } \right) d y = 0y2dx−(xy+x2)dy=0

The General Solution Of $y ^ { 2 } d x - \left( x y + x ^ { 2 } \right) d y = 0$