Linear ODEs

Introduction to Differential Equations: Basics of differential equations and its types like ordinary and partial differential equations.

First-Order Differential Equations: Solution techniques of first-order linear and non-linear differential equations, like exact, separable, and homogeneous types.

Second-Order Differential Equations: Linear second-order differential equations and their basic properties like solutions, existence, and uniqueness.

Homogeneous Linear Differential Equations: Homogeneous differential equations, including linear differential equations, non-linear differential equations, and their solutions.

Non-Homogeneous Linear Differential Equations: Non-homogeneous linear differential equations, their solutions, and applications.

Higher-Order Linear Differential Equations: Higher-order linear differential equations of order n with constant coefficients and their solutions.

Systems of Differential Equations: Systems of first-order linear differential equations and their solutions.

Laplace Transforms: Introduction to Laplace transform, its definition, properties, and Laplace transforms of common functions.

Fourier Series: Introduction to Fourier series, its definition, properties and Fourier series of common functions.

Numerical Methods: Euler's method, Runge-Kutta methods, finite difference methods, numerical solutions of differential equations, and truncation errors.

Stability Analysis: Stability of linear differential equations, equilibrium points, stability analysis through phase spaces.

Applications: Differential equations applications in various fields like engineering, physics, finance, economics, etc.

First-order linear differential equation: An equation of the form y'+p(x)y=q(x) where p and q are functions of x.

Second-order homogeneous linear differential equation: An equation of the form y''+ay'+by=0 where a and b are constants.

Second-order non-homogeneous linear differential equation: An equation of the form y''+ay'+by=f(x) where a, b, and f are functions of x.

Constant-coefficient linear differential equation: An equation of the form y^(n)+a_(n-1)y^(n-1)+...+a_0y=f(x) where a_0, a_1,...,a_(n-1) are constants.

Nth-order homogeneous linear differential equation: An equation of the form y^(n)+a_(n-1)y^(n-1)+...+a_0y=0 where a_0, ...,a_(n-1) are constants.

Nth-order non-homogeneous linear differential equation: An equation of the form y^(n)+a_(n-1)y^(n-1)+...+a_0y=f(x) where a_0, ...,a_(n-1) are constants and f is a function of x.

Linear differential equation with variable coefficients: An equation of the form y''+p(x)y'+q(x)y=r(x) where p(x), q(x), and r(x) are functions of x.

Homogeneous linear differential equation with constant coefficients: An equation of the form ay''+by'+cy=0 where a,b, and c are constants.

Non-homogeneous linear differential equation with constant coefficients: An equation of the form ay''+by'+cy=f(x) where a,b, and c are constants.

Cauchy-Euler differential equation: A differential equation of the form x^2y''+pxy'+qy=0.