Question

$\frac{dy}{dx}=e^{x+y}+e^y{x}^3$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}=e^{x+y}+e^y{x}^3 \\ \Rightarrow \frac{dy}{dx}=e^x{e}^y+e^y{x}^3 \\ \Rightarrow \frac{dy}{dx}=e^y\left(e^x+x^3\right) \\ \Rightarrow \left(e^x+x^3\right)dx=\frac{1}{e^y}dy \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \left(e^x+x^3\right)dx=\int \frac{1}{e^y}dy \\ \Rightarrow e^x+\frac{x^4}{4}=-{\mathrm{e}}^{-y}+C \\ \Rightarrow {\mathrm{e}}^x+{\mathrm{e}}^{-\mathrm{y}}+\frac{x^4}{4}=C \\ \mathrm{Hence},{\mathrm{e}}^x+{\mathrm{e}}^{-\mathrm{y}}+\frac{x^4}{4}=C\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$