Nonlinear Differential Equation

A Differential Equation is nonlinear if terms don’t appear only as

$$f(t)\cdot y^{(n)}(t)$$

Note that $y^{(n)}(t)$ is the $n^{\text{th}}$ derivative of $y(t)$ in Lagrange notation.

Linear	Nonlinear
$y^{\prime }+2y=0$	$y^{\prime }+y^2=0$
$y^{\prime }+2y=t^2$	$y^{\prime }+y^2=t^2$
$y^{\prime }+t^{3}y=0$	$y^{\prime }+t{y}^3=0$
$t^2{y}^{\prime \prime \prime }-\cos (t)y^{\prime \prime }+y=0$	$y^{\prime \prime }+3y{y}^{\prime }=7\cos (t)$

First Order

$$\frac{dy}{dt}=f(t,y)$$

Solution Method No guaranteed general solution method. However, Separation of Variables is sometimes possible. General Solution Will have a free parameter $C$