Q.Solve: $y(1 + xy)\,dx - x\,dy = 0$.

Nonlinear Differential Equations

The idea

A differential equation tells you how a quantity changes. It is called linear when the unknown function and its derivatives appear only to the first power and are never multiplied together — that is the clean, well-behaved case NCERT's methods (variable-separable, homogeneous, linear first-order) are built around. A nonlinear differential equation is simply any equation that breaks this rule.

The precise statement

A differential equation is linear if it can be written as

where the coefficients $a_i(x)$ and $f(x)$ depend only on the independent variable $x$ (or are constants), and $y$ together with its derivatives appears only to the first power and is never multiplied together.

A nonlinear differential equation violates at least one of these conditions:

• a derivative or $y$ raised to a power other than $1$, e.g. $y^2$ or $\left(\dfrac{dy}{dx}\right)^{2}$;
• a product of $y$ with one of its derivatives, e.g. $y\dfrac{dy}{dx}$;
• $y$ appearing inside a nonlinear function, e.g. $\sin y$, $e^{y}$, $\log y$.

Examples

Linear (for contrast):

Nonlinear:

Why the distinction matters …