Solve the differential equation.
-$$x ^ { 2 } \frac { d y } { d x } = 3 y$$
A) $\mathrm { y } = \mathrm { Ce } ^ { 3 / \mathrm { x } }$
B) $y = C e ^ { - 3 / x }$
C) $y = - \frac { 3 } { x } + C$
D) $\mathrm { y } = \mathrm { Ce } ^ { - 3 \mathrm { x } }$

Question

Quizplus · Accepted Answer

First, let's rearrange the equation:
$$x^2 \frac{dy}{dx} = 3y$$
$$\frac{dy}{y} = \frac{3}{x^2} dx$$
Integrate both sides:
$$\ln|y| = -\frac{3}{x}+C$$
$$|y| = e^{-\frac{3}{x}+C} = e^C e^{-\frac{3}{x}}$$
Note that we can replace the absolute value with a constant since both $y$ and $-y$ are solutions. Let $C' = e^C$, then we have:
$$y = C'e^{-\frac{3}{x}}$$
which matches choice (B). Thus, the answer is (B).

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