Question

General solution of the differential equation \[ \frac{dy}{dx}=e^{x-y}+3x^{2}e^{-y} \] is:

• A. \(e^{-y}+e^{x}=x^{3}+c \)
• B. \(e^{-y}=e^{x}+x^{3}+c \)
• C. \(e^{y}=e^{x}-x^{3}+c \)
• D. \(e^{y}=e^{x}+x^{3}+c \)

Hint:

Whenever you see $e^{x-y}$ or $e^{-y}$ together, try factoring $e^{-y}$ first to separate variables easily.

Answer 1

The Correct Option is B

Concept: This type of differential equation is solved using substitution and separation. When terms contain $e^{-y}$, we try to rewrite the equation in terms of $e^{y}$ or $e^{-y}$ to make it separable.

Step 1: {Rewrite the given equation.}
\[ \frac{dy}{dx}=e^{x-y}+3x^{2}e^{-y} \]

Step 2: {Factor out $e^{-y}$.}
\[ \frac{dy}{dx}=e^{-y}(e^{x}+3x^{2}) \]

Step 3: {Multiply both sides by $e^{y}$.}
\[ e^{y}\frac{dy}{dx}=e^{x}+3x^{2} \]

Step 4: {Rewrite in differential form.}
\[ e^{y}dy=(e^{x}+3x^{2})dx \]

Step 5: {Integrate both sides.}
\[ \int e^{y}dy=\int (e^{x}+3x^{2})dx \]

Step 6: {Compute integrals.}
\[ e^{y}=e^{x}+x^{3}+c \]