Question

(1 + x) y dx + (1 + y) x dy = 0

Answer 1

We have,
(1 + x) y dx + (1 + y) x dy = 0

$$\begin{gathered} \frac{dy}{dx}=-\frac{y\left(1+x\right)}{x\left(1+y\right)} \\ \Rightarrow \left(\frac{1+y}{y}\right)dy=-\left(\frac{1+x}{x}\right)dx \\ \Rightarrow \left(\frac{1}{y}+y\right)dy=-\left(\frac{1}{x}+1\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \left(\frac{1}{y}+1\right)dy=-\int \left(\frac{1}{x}+1\right)dx \\ \Rightarrow \int \frac{1}{y}dy+\int dy=-\int \frac{1}{x}dx-\int dx \\ \Rightarrow \log \left|y\right|+y=-\log \left|x\right|-x+C \\ \Rightarrow \log \left|xy\right|+y+x=C \\ \Rightarrow x+y+\log \left|xy\right|=C \end{gathered}$$