Question

(xy² + 2x) dx + (x² y + 2y) dy = 0

Answer 1

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