"In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives."
Mathematical models used to describe physical phenomena and changes over time involving rates of change.
Ordinary Differential Equations (ODEs): These are equations that contain derivatives of a single variable with respect to another variable. They are the most basic type of differential equation and are used to model a wide variety of physical phenomena.
Partial Differential Equations (PDEs): These are equations that contain partial derivatives of two or more variables. They are used to model more complex physical phenomena, such as fluid flow, electromagnetic fields, and heat transfer.
Linear Differential Equations: These are equations that can be written in a linear form, where each term is either a constant or a linear combination of the dependent variable and its derivatives. They have a closed-form solution and are easier to solve than nonlinear differential equations.
Nonlinear Differential Equations: These are equations that cannot be expressed in a linear form. They are more difficult to solve than linear differential equations and often require numerical methods to obtain a solution.
Boundary Value Problems: These are differential equations that are subject to boundary conditions, which specify the values of the dependent variable or its derivatives at the boundaries of the solution domain. The solution to a boundary value problem is unique and requires the use of boundary value techniques.
Initial Value Problems: These are differential equations that are subject to an initial condition, which specifies the value of the dependent variable at a given point. The solution to an initial value problem is also unique but requires the use of initial value techniques.
Homogeneous Differential Equations: These are equations that contain only the dependent variable and its derivatives. They can be solved by assuming a specific form for the solution and then determining the constants that satisfy the boundary or initial conditions.
Inhomogeneous Differential Equations: These are equations that contain a forcing term, which is a function of the independent variable. They require the use of a particular solution, which is added to the general solution of the homogeneous equation.
Separable Differential Equations: These are equations that can be separated into two simple equations, one containing only the dependent variable and the other containing only the independent variable. They are relatively easy to solve and are commonly used to model population growth, radioactive decay, and chemical reactions.
Exact Differential Equations: These are equations that can be expressed as the total derivative of a function. They can be solved by finding this function and then integrating it with respect to the independent variable.
Series Solutions: These are differential equations that cannot be solved analytically but can be approximated by a power series. The coefficients of the series are determined by a recurrence relation and the solution is obtained by summing the series.
Laplace Transform: This is a mathematical tool used to transform a differential equation into an algebraic equation. It is particularly useful for solving linear differential equations with constant coefficients and for solving boundary value problems.
Fourier Series: This is a mathematical series that represents a periodic function as a sum of sine and cosine functions. It is used to solve boundary value problems involving periodic functions.
Numerical Methods: These are methods used to approximate the solution of a differential equation when an analytical solution is not possible or too complex. Common numerical methods include Euler's method, Runge-Kutta method, and finite difference method.
Stability Analysis: This is the study of the behavior of solutions to differential equations over time. It is used to determine whether a system is stable or unstable and to predict its long-term behavior.
Ordinary Differential Equations (ODEs): These are equations that involve only one independent variable and one or more dependent variables with their derivatives.
Partial Differential Equations (PDEs): These are equations that involve two or more independent variables and one or more dependent variables with their partial derivatives.
Linear Differential Equations: These are differential equations where the dependent variable and its derivatives appear linearly.
Nonlinear Differential Equations: These are differential equations where the dependent variable and its derivatives appear nonlinearly.
Homogeneous Differential Equations: These are differential equations where the terms containing the dependent variable and its derivatives have homogeneous dimensions.
Non-Homogeneous Differential Equations: These are differential equations where the terms containing the dependent variable and its derivatives have non-homogeneous dimensions.
Autonomous Differential Equations: These are differential equations where the independent variable does not explicitly appear.
Stochastic Differential Equations: These are differential equations that involve random variables and the solutions are random processes.
First Order Differential Equations: These are differential equations that involve only first-order derivatives.
Second Order Differential Equations: These are differential equations that involve second-order derivatives.
Higher-Order Differential Equations: These are differential equations of order higher than two.
Separable Differential Equations: These are differential equations that can be separated into two functions of the dependent and independent variables.
Exact Differential Equations: These are differential equations that can be expressed as total derivatives of an unknown function.
Inexact Differential Equations: These are differential equations that cannot be expressed as total derivatives of an unknown function.
Bernoulli Differential Equations: These are nonlinear differential equations that can be transformed into linear equations.
Riccati Differential Equations: These are nonlinear differential equations that can be transformed into linear equations.
The Euler-Lagrange Equation: It is a PDE that appears in the calculus of variations.
The Laplace Equation: It is a partial differential equation arising in electrostatics, fluid dynamics, and other fields.
The Heat Equation: It is a partial differential equation arising in diffusion, heat transfer, and other fields.
The Wave Equation: It is a partial differential equation arising in wave phenomena in physics and other fields.
"The functions generally represent physical quantities, the derivatives represent their rates of change."
"Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology."
"The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions."
"Only the simplest differential equations are soluble by explicit formulas."
"Many properties of solutions of a given differential equation may be determined without computing them exactly."
"When a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers."
"The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations."
"Many numerical methods have been developed to determine solutions with a given degree of accuracy."
"Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology."
"The functions generally represent physical quantities, the derivatives represent their rates of change."
"The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions."
"Only the simplest differential equations are soluble by explicit formulas."
"Many properties of solutions of a given differential equation may be determined without computing them exactly."
"When a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers."
"The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations."
"Many numerical methods have been developed to determine solutions with a given degree of accuracy."
"Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology."
"A differential equation is an equation that relates one or more unknown functions and their derivatives."
"Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology."