Question

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

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}=e^{x+y}+x^2{e}^y \\ \Rightarrow \frac{dy}{dx}=e^x{e}^y+x^2{e}^y \\ \Rightarrow \frac{dy}{dx}=e^y\left(e^x+x^2\right) \\ \Rightarrow \frac{1}{e^y}dy=\left(e^x+x^2\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{e^y}dy=\int \left(e^x+x^2\right)dx \\ \Rightarrow -e^{-y}=e^x+\frac{x^3}{3}+\mathrm{C} \end{gathered}$$