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Single Variable Calculus

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Moderate

Separable Differential Equation Involving Exponential Function

SVCALC-BWGK8N

Solve the following differential equation where $y(0)=\frac{1}{2}$ and determine the interval of validity for the solution.

$$ \frac{dy}{dx}=\frac{5 x^2}{e^y}$$

A

$ e^y =\dfrac{5}{3} x^3 +C,\, x \mbox{ all real numbers}$

B

$ y = \ln\left (\dfrac{5}{3} x^3 + \sqrt{e}\right), \, x >- \sqrt[3]{ \dfrac{3 e^{1/2}}{5}}$

C

$ y = \ln\left(\dfrac{5}{3} x^3 \right), \, x > \left(- \frac{3 e^{1/2}}{5}\right)^{1/3}$

D

$ e^y =\dfrac{5}{3} x^3 +e^{1/2}, \, x \mbox{ all real numbers}$