Question

(1 + y + x² y) dx + (x + x³) dy = 0

Answer 1

$$\begin{gathered} \left(1+y+x^{2}y\right)dx+\left(x+x^3\right)dy=0 \\ \Rightarrow dx+y\left(1+x^2\right)dx+x\left(1+x^2\right)dy=0 \\ \Rightarrow dx+\left(1+x^2\right)\left[ydx+xdy\right]=0 \\ \Rightarrow \left(1+x^2\right)\left[ydx+xdy\right]=-dx \\ \Rightarrow \left[ydx+xdy\right]=-\frac{1}{\left(1+x^2\right)}dx \\ \Rightarrow \left[ydx+xdy\right]=-\frac{dx}{\left(1+x^2\right)} \end{gathered}$$

On integrating both side we get,

$$\begin{gathered} \left(xy\right)=-\int \frac{1}{1+x^2}dx \\ \Rightarrow xy=-{\tan }^{-1}x+c \\ \Rightarrow xy+{\tan }^{-1}x=c \end{gathered}$$