Question

(xy² + x) dx + (y − x²y) dy = 0

Answer 1

\[\left( x y^2 + x \right)dx + \left( y - x^2 y \right)dy = 0\]
\[ \Rightarrow x\left( y^2 + 1 \right)dx = y\left( x^2 - 1 \right)dy\]
\[ \Rightarrow \frac{x\left( y^2 + 1 \right)}{y\left( x^2 - 1 \right)} = \frac{dy}{dx}\]
\[ \Rightarrow x\left( y^2 + 1 \right)\frac{dy}{dx} - y\left( x^2 - 1 \right) = 0\]
\[ \Rightarrow \left( y^2 + 1 \right)\frac{dy}{dx} - y\left( x - \frac{1}{x} \right) = 0\]
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.