Definition of a Linear Differential Equation

Definition

A differential Equation where the dependent variable is linear in called a Linear Differential Equation.

In general

$$y^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=f(t)$$

Where, $y\to$ is linear in every derivative and $a_i(t)$ is arbitrary function dependent on the particular equation under study that don’t have to be linear.

Example

1. Linear Differential Equation

$$y^{\prime \prime }+y=0$$

2. Nonlinear Differential Equation:

$$y^{\prime \prime }+y^2=0$$