The general solution of the first-order differential equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$ is

A) $4 x y = e ^ { x } + C e ^ { - x }$
B) $3 x y = e ^ { x } - C e ^ { - x }$
C) $3 x y = e ^ { x } + C e ^ { - x }$
D) $2 x y = e ^ { x } - C e ^ { - x }$
E) $2 x y = e ^ { x } + C e ^ { - x }$

Question

The general solution of the first-order differential equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$ is

A) $4 x y = e ^ { x } + C e ^ { - x }$
B) $3 x y = e ^ { x } - C e ^ { - x }$
C) $3 x y = e ^ { x } + C e ^ { - x }$
D) $2 x y = e ^ { x } - C e ^ { - x }$
E) $2 x y = e ^ { x } + C e ^ { - x }$

Quizplus · Accepted Answer

This differential equation can be solved using an integrating factor. The integrating factor is $e^{\int \frac{1+x}{x} dx} = e^{\ln|x| + x} = xe^x$. Multiplying both sides of the equation by this integrating factor transforms it into an exact differential, which can be integrated directly to find the solution. The correct solution, after simplification, is $2xy = e^x + Ce^{-x}$, which matches option E.

The General Solution of the First-Order Differential Equation xy′+(1+x)y=exx y ^ { \prime } + ( 1 + x ) y = e ^ { x }xy′+(1+x)y=ex

The General Solution of the First-Order Differential Equation $x y ^ { \prime } + ( 1 + x ) y = e ^ { x }$