Question

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

Answer 1

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