Solve the differential equation.
$$y ^ { \prime } = x e ^ { - \sin x } - y \cos x$$
A) $y = \left( \frac { 1 } { 2 } x ^ { 2 } + C \right) e ^ { - \sin x }$
B)
$y = \frac { 1 } { 2 } x + C e ^ { - \cos x }$
C) $y = C e ^ { - \sin x }$
D) $y = e ^ { - \sin x } + C x$
E) $y = \left( 2 x ^ { 2 } + C \right) e ^ { - x } \sin x$

Question

Quizplus · Accepted Answer

The given differential equation is a first-order linear differential equation. The integrating factor is $e^{\int \cos x \, dx} = e^{\sin x}$. Multiplying through by the integrating factor and solving gives the solution $y = \left( \frac{1}{2}x^2 + C \right) e^{-\sin x}$.

Solve the Differential Equation y′=xe−sin⁡x−ycos⁡xy ^ { \prime } = x e ^ { - \sin x } - y \cos xy′=xe−sinx−ycosx

Solve the Differential Equation $y ^ { \prime } = x e ^ { - \sin x } - y \cos x$