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

This differential equation can be solved by recognizing it as a first-order linear differential equation in the form $x^2 \frac{dy}{dx} - y = g(x)$, where $g(x) = 2x^3e^{-1/x}$. The integrating factor is $e^{\int -1/x \, dx} = e^{-\ln|x|} = \frac{1}{x}$. Multiplying through by this integrating factor transforms the equation into a form where the left side is the derivative of a product, leading to the solution $y = x^2e^{-1/x} + Ce^{-1/x}$.

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