Question

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

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ dy+\left(x+1\right)\left(y+1\right)dx=0 \\ \Rightarrow dy=-\left(x+1\right)\left(y+1\right)dx \\ \Rightarrow \frac{1}{y+1}dy=-\left(x+1\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{y+1}dy=-\int \left(x+1\right)dx \\ \Rightarrow \log \left|y+1\right|=-\frac{x^2}{2}-x+\mathrm{C} \\ \Rightarrow \log \left|y+1\right|+\frac{x^2}{2}+x=\mathrm{C} \\ \mathrm{Hence},\log \left|y+1\right|+\frac{x^2}{2}+x=\mathrm{C}\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$