The study of equations involving derivatives, including techniques for solving and modeling dynamic systems.
Ordinary Differential Equations (ODEs): ODEs are equations that relate a function and its derivatives. They are solved for a specific function with respect to one variable.
Partial Differential Equations (PDEs): PDEs are equations that relate a function and its partial derivatives. They are solved for a specific function with respect to multiple variables.
First-order ODEs: ODEs that involve only first-order derivatives.
Second-order ODEs: ODEs that involve only second-order derivatives.
Higher-order ODEs: ODEs that involve derivatives of order higher than second.
Homogeneous ODEs: ODEs where every term is of the same degree in the dependent variable and its derivatives.
Non-homogeneous ODEs: ODEs where the dependent variable is not proportional to its derivatives and the equation is not homogeneous.
Initial value problems (IVPs): ODEs where the initial values of the dependent variable and its derivatives are given.
Boundary value problems (BVPs): ODEs where the values of the dependent variable are given at two or more distinct points.
Existence and uniqueness of solutions: Theorems that guarantee the existence and uniqueness of solutions to certain types of ODEs.
Separable ODEs: ODEs where the dependent variable can be separated from its derivative terms, allowing them to be solved analytically.
Linear ODEs: ODEs that can be expressed as a linear combination of the dependent variable and its derivatives.
Exact ODEs: ODEs that can be solved using an integrating factor to make the left-hand side of the equation an exact differential.
Integrating factors: Functions used to transform a non-exact ODE into an exact one.
Method of undetermined coefficients: A technique used to find a particular solution to a non-homogeneous linear ODE.
Variation of parameters: A technique used to find the general solution to a non-homogeneous linear ODE.
Sturm-Liouville problems: A special type of BVP that involves second-order linear ODEs and the associated eigenvalue problem.
Fourier series: A way to express periodic functions as an infinite sum of sines and cosines.
Heat equation: A PDE that models the diffusion of heat through a solid material.
Wave equation: A PDE that models the propagation of waves through a medium.
Laplace equation: A PDE that arises in many areas of physics and engineering, including fluid mechanics and electrostatics.
Finite difference methods: Numerical methods used to discretize PDEs and solve them on a computer.