Question

xy (y + 1) dy = (x² + 1) dx

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ xy\left(y+1\right)dy=\left(x^2+1\right)dx \\ \Rightarrow \left\{y\left(y+1\right)\right\}dy=\frac{x^2+1}{x}dx \\ \Rightarrow \left(y^2+y\right)dy=\left(x+\frac{1}{x}\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \left(y^2+y\right)dy=\int \left(x+\frac{1}{x}\right)dx \\ \Rightarrow \int y^{2}dy+\int ydy=\int xdx+\int \frac{1}{x}dx \\ \Rightarrow \frac{y^3}{3}+\frac{y^2}{2}=\frac{x^2}{2}+\log \left|x\right|+C \\ \mathrm{Hence},\frac{y^3}{3}+\frac{y^2}{2}=\frac{x^2}{2}+\log \left|x\right|+C\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$