# Linear Differential Equations

### Linear Differential Equations:

1. 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 ${a}_{n}\left(x\right)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n-1}\left(x\right)\frac{{d}^{n-1}y}{d{x}^{n-1}}+\cdots +{a}_{1}\left(x\right)\frac{dy}{dx}+{a}_{0}\left(x\right)y=g\left(x\right)$
2. Types:

a. Homogeneous Linear Differential Equations:

• Equations where the right-hand side $g\left(x\right)$ is zero.
• Example: ${a}_{n}\left(x\right)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n-1}\left(x\right)\frac{{d}^{n-1}y}{d{x}^{n-1}}+\cdots +{a}_{1}\left(x\right)\frac{dy}{dx}+{a}_{0}\left(x\right)y=0$

b. Non-Homogeneous Linear Differential Equations:

• Equations where the right-hand side $g\left(x\right)$ is non-zero.
• Example: ${a}_{n}\left(x\right)\frac{{d}^{n}y}{d{x}^{n}}+{a}_{n-1}\left(x\right)\frac{{d}^{n-1}y}{d{x}^{n-1}}+\cdots +{a}_{1}\left(x\right)\frac{dy}{dx}+{a}_{0}\left(x\right)y=g\left(x\right)$.
3. 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. Non-Homogeneous 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.
4. 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.
5. 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.
6. Importance:

• Linear differential equations are extensively studied and widely applicable due to their solvability and relevance in modeling various real-world systems.