Solve the differential equation.
$$x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$$
A) $y = C x e ^ { 2 x }$
B) $y = C e ^ { - 2 / x }$
C) $y = e ^ { 1 / x } x + C e ^ { 1 / x }$
D) $y = x ^ { 2 } e ^ { - 1 / x } + C e ^ { - 1 / x }$
E) none of these

Question

Quizplus · Accepted Answer

We start by dividing both sides by $x^2$ which gives
$$\frac{dy}{dx} - \frac{y}{x^2} = 2xe^{-1/x}.$$
The left hand side is now in the form $\frac{d}{dx}\left(\frac{y}{x}\right),$ so we integrate both sides with respect to $x$ to obtain
$$\frac{y}{x} = \int 2xe^{-1/x}\,dx = -x(e^{-1/x})'+C.$$
Simplifying this expression, we get
$$y = x^2e^{-1/x}+Cx.$$
Therefore, the answer is D.

Solve the Differential Equation x2dydx−y=2x3e−1/xx ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }x2dxdy​−y=2x3e−1/x

Solve the Differential Equation $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$