Linear Differential Equations
Linear Differential Equations:

Definition:
 A linear differential equation is an equation in which the dependent variable and its derivatives appear as linear terms, not raised to any power other than 1.
 The general form of a linear differential equation of order $n$ is ${a}_{n}(x)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n1}(x)\frac{{d}^{n1}y}{d{x}^{n1}}+\cdots +{a}_{1}(x)\frac{dy}{dx}+{a}_{0}(x)y=g(x)$

Types:
a. Homogeneous Linear Differential Equations:
 Equations where the righthand side $g(x)$ is zero.
 Example: ${a}_{n}(x)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n1}(x)\frac{{d}^{n1}y}{d{x}^{n1}}+\cdots +{a}_{1}(x)\frac{dy}{dx}+{a}_{0}(x)y=0$
b. NonHomogeneous Linear Differential Equations:
 Equations where the righthand side $g(x)$ is nonzero.
 Example: ${a}_{n}(x)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n1}(x)\frac{{d}^{n1}y}{d{x}^{n1}}+\cdots +{a}_{1}(x)\frac{dy}{dx}+{a}_{0}(x)y=g(x)$.

Solution Methods:
a. Homogeneous Linear Equations:
 Solve using the characteristic equation method or by assuming solutions of the form $y={e}^{rx}$ and substituting into the equation.
b. NonHomogeneous Linear Equations:
 Solve the associated homogeneous equation to find the complementary function (CF).
 Use particular integrals (PI) or method of undetermined coefficients to find the particular solution (PS).
 The general solution is the sum of CF and PS.

Properties:
a. Superposition Principle:
 The sum of any two solutions of a linear differential equation is also a solution.
 The general solution is a linear combination of individual solutions.
b. Linearity:
 The equation is linear in terms of the dependent variable and its derivatives.

Applications:
 Used extensively in physics, engineering, and mathematical modeling to describe systems with linear behavior.
 Examples include harmonic oscillators, circuits, population dynamics, and many physical phenomena.

Importance:
 Linear differential equations are extensively studied and widely applicable due to their solvability and relevance in modeling various realworld systems.