Linear Differential Equations

The general linear differential equation of order n has the form
$\displaystyle a_0(x)\frac{d^ny}{dx^n} + a_1(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{n-1}(x)\frac{dy}{dx} + a_n(x)y = R(x)\ \ \ (1)$
A differential equation which cannot be written in this form is called nonlinear.

$\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = \sin x$ is a second order linear equation. $\displaystyle y\frac{d^2y}{dx^2} - x\left(\frac{dy}{dx}\right)^2 + x^2y = e^{-x}$ is a second order nonlinear equation.

If R(x), the right side of (1), is replaced by zero the resulting equation is called the complementary, reduced or homogeneous equation. If R(x) ≠ 0, the equation is called the complete or nonhomogeneous equation.

If $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -2xy =\sin x$ is the complete equation, then $\displaystyle x\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2xy = 0$ is the corresponding complementary, reduced or homogeneous equation.

If $a_0(x)\, \cdots \,a_n(x)$ are all constants, (1) is said to have constant coefficient, otherwise it is said to have variable coefficients.