Topic: Mathematics
The solution of differential equation $dy/dx=e^{x-y}+x^{2}e^{-y}$ is
- A. $e^{y}=e^{x}+x^{3}/3+c$
- B. $e^{y}-e^{x}=c$
- C. $x-e^{y}=c$
- D. $e^{y}+e^{x}+x^{3}/3+y=0$
Answer: A. $e^{y}=e^{x}+x^{3}/3+c$
Explanation: $\frac{dy}{dx}=e^{x-y}+x^{2}e^{-y} \Rightarrow \frac{dy}{dx}=\frac{e^{x}+x^{2}}{e^{y}} \Rightarrow e^{y}dy=(e^{x}+x^{2})dx$. Integrating both sides, $\int e^{y}dy=\int e^{x}dx+\int x^{2}dx \Rightarrow e^{y}=e^{x}+\frac{x^{3}}{3}+c$.