"In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives."
A differential equation is a mathematical equation that relates some function with its derivatives.
Introduction and Basics: This topic introduces the concept of differential equations, their types, and how to write them. It also covers basic concepts like order, degree, and linearity.
Separable Equations: Separable equations are the ones that can be separated into two functions, each containing only one variable. This topic deals with finding their general solutions.
First-Order Linear Equations: This topic covers linear differential equations, which are the ones that can be expressed in the form of a linear function of the dependent variable and its derivative.
Homogeneous Equations: This topic deals with homogeneous differential equations, which are the ones where the coefficients of the equation are functions of only one variable.
Exact Equations: This topic explains exact differential equations, which are the ones where the left and right-hand sides of the equation differ by an exact differential.
Bernoulli Equations: This topic deals with Bernoulli differential equations, which are the ones in which the dependent variable appears in both linear and nonlinear terms.
Second-Order Linear Equations: This topic covers second-order linear differential equations, which involve second-order derivatives of the dependent variable.
Reduction of Order: This topic explains the reduction of order method, which is used to solve second-order linear equations by reducing them to first-order equations.
Fundamental Solution Sets: This topic deals with the concept of the fundamental solution set, which is a set of solutions that can be used to express the general solution of a homogeneous linear equation.
Variation of Parameters: This topic explains the variation of parameter method, which is a technique for finding the general solution of a non-homogeneous linear equation.
Laplace Transform: This topic covers the Laplace transform, which is a mathematical tool used to solve differential equations.
Systems of Differential Equations: This topic deals with systems of differential equations, which are the ones that involve two or more dependent variables.
Numerical Solutions: This topic covers the numerical methods for solving differential equations, including Euler's method and the Runge-Kutta method.
Ordinary Differential Equations (ODEs): ODEs deal with functions of a single variable and their derivatives or differentials.
Partial Differential Equations (PDEs): PDEs deal with functions of several variables and their partial derivatives.
First-Order Differential Equations: These are ODEs that involve only the first derivative of the function.
Second-Order Differential Equations: These are ODEs that involve the second derivative of the function.
Linear Differential Equations: These are ODEs or PDEs that involve linear combinations of the function and its derivatives.
Nonlinear Differential Equations: These are ODEs or PDEs that involve products, powers, or other nonlinear operations of the function and its derivatives.
Homogeneous Differential Equations: These are ODEs or PDEs where all the terms involving the function and its derivatives have the same degree.
Inhomogeneous Differential Equations: These are ODEs or PDEs where some terms involving the function and its derivatives have different degrees.
Autonomous Differential Equations: These are ODEs where the independent variable does not explicitly appear.
Separable Differential Equations: These are ODEs that can be separated into two parts, one involving only the function and one involving only the independent variable.
"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."