Question

(xy² + x) dx + (y − x²y) dy = 0

Answer 1

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

In this differential equation, the order of the highest order derivative is 1 and its power is 1. So, it is a differential equation of degree 1 and order 1.

It is a non-linear equation, as the product containing dependent variable and its differential co-efficient $\left(y^2\frac{dy}{dx}\right)$ is present in it.