Solve the differential equation.
-$$\frac { d y } { d x } = 3 x ^ { 2 } e ^ { - y }$$
A) $y = \ln \left( x ^ { 3 } + C \right)$
B) $y = \ln \left( 3 x ^ { 3 } + C \right)$
C) $y = C \ln \left( x ^ { 3 } \right)$
D) $y = x ^ { 3 } + C$

Question

Quizplus · Accepted Answer

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We notice that the differential equation is separable:
$$\frac { d y } { d x } = 3 x ^ { 2 } e ^ { - y } \Rightarrow e^y dy=3x^2dx$$
Integrating both sides gives:
$$e^y = x^3 + C$$
Taking $\ln$ of both sides, we get:
$$y = \ln \left( x ^ { 3 } + C \right)$$
Thus, the answer is choice A.

Solve the Differential Equation dydx=3x2e−y\frac { d y } { d x } = 3 x ^ { 2 } e ^ { - y }dxdy​=3x2e−y

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