The solution of the differential equation $y ^ { \prime } + y = x$ is
A) $y = x - 1 + c e ^ { - x }$
B) $y = x ^ { 2 } / 2 + e ^ { x }$
C) $y = x ^ { 2 } / 2 + e ^ { - x }$
D) $y = x - 1 + c e ^ { x }$
E) $y = - x - 1 + c e ^ { x }$

Question

Quizplus · Accepted Answer

The given differential equation is a first-order linear differential equation. The solution to such an equation can be found using the integrating factor method. The integrating factor is $e^{\int 1 dx} = e^x$. Multiplying both sides of the original equation by this integrating factor and integrating yields the solution $y = x - 1 + c e^{-x}$, where $c$ is the constant of integration.

The Solution of the Differential Equation y′+y=xy ^ { \prime } + y = xy′+y=x

The Solution of the Differential Equation $y ^ { \prime } + y = x$