The solution of the differential equation d y/d x=e^{x-y}+x² e^{-y} is...

Correct Answer is : ey=ex+x33+ce^{y}=e^{x}+{x³}{3}+cey=ex+3x3+c

Correct Answer is : ey=ex+x33+ce^{y}=e^{x}+{x³}{3}+cey=ex+3x3+c

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MHTCET 15 Apr 2026 Shift 1 Paper

Question : 135 of 150

Marks: +1, -0

\frac{d y}{d x}=e^{x-y}+x^{2} e^{-y}

is...

Solution:

Concept:

Rewrite the given differential equation by separating terms and integrating after using the chain rule.

Explanation:

The given equation is

\frac{d y}{d x}=e^{x-y}+x^{2} e^{-y}

Using

e^{x-y}=e^{x} e^{-y}

, factor out

e^{-y}

\frac{d y}{d x}=e^{-y}\left(e^{x}+x^{2}\right)

Multiply both sides by

e^{y}

e^{y} \frac{d y}{d x}=e^{x}+x^{2}

Since

\frac{d}{d x}\left(e^{y}\right)=e^{y} \frac{d y}{d x}

, we get

\frac{d}{d x}\left(e^{y}\right)=e^{x}+x^{2}

Integrate both sides with respect to

x

e^{y}=\int\left(e^{x}+x^{2}\right) d x

e^{y}=e^{x}+\frac{x^{3}}{3}+c

Answer:

e^{y}=e^{x}+\frac{x^{3}}{3}+c

Correct option: Option C.

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