# Differential Equation

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Mainly the study of differential equations consists 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 solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often 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, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

from Differential equation - Wikipedia

This topic includes the following resources and journeys:

Filters
17 items

## Numerically Calculating Partial Derivatives

59 min
Beginner
Video
Theory

In this video we discuss how to calculate partial derivatives of a function using numerical techniques. In other words, these partials are calculated withou...

See More

## Derivation and Solution of Laplace’s Equation

33 min
Beginner
Video
Theory

In this video we show how the heat equation can be simplified to obtain Laplace’s equation. We investigate how to solve Laplace’s equation using separation ...

See More

## Derivation of the 1D Wave Equation

26 min
Beginner
Video
Theory

In this video, we derive the 1D wave equation. This partial differential equation (PDE) applies to scenarios such as the vibrations of a continuous string. ...

See More

## Homogeneous Linear Ordinary Differential Equations

74 min
Beginner
Video
Theory

In this video we discuss how to solve homogeneous linear ordinary differential equations (ODEs). The approach outlined in this lecture is applicable to high...

See More

## Derivation of the 2D Wave Equation

27 min
Beginner
Video
Theory

In this video we derive the 2D wave equation. This partial differential equation governs the motion of waves in a plane and is applicable for thin vibrating...

See More

## Heat Transfer Demonstration

63 min
Beginner
Video
Theory

In this video we demonstrate heat transfer through a metal bar. By heating one side of the bar we can impose a non-uniform temperature distribution across t...

See More

## Nonhomogeneous Linear Ordinary Differential Equations

70 min
Beginner
Video
Theory

In the previous video (https://youtu.be/3Kox-3APznI) we examined solving homogeneous linear ordinary differential equations (the forcing function was equal t...

See More

## Solving the 2D Wave Equation

73 min
Beginner
Video
Theory

In this video, we solve the 2D wave equation. We utilize two successive separation of variables to solve this partial differential equation. Topics discuss...

See More

## Solving the 1D Wave Equation

118 min
Beginner
Video
Theory

In this video, we solve the 1D wave equation. We utilize the separation of variables method to solve this 2nd order, linear, homogeneous, partial differenti...

See More

## Introduction to Ordinary Differential Equations

35 min
Beginner
Video
Theory

In this video we introduce the concept of ordinary differential equations (ODEs). We give examples of how these appear in science and engineering as well as...

See More

## Partial Fraction Expansion/Decomposition

59 min
Beginner
Video
Theory

In this video we discuss how to perform partial fraction expansion (PFE) to rewrite a ratio of polynomials as simpler expressions. Topics and time stamps:(0...

See More

## Derivation of the Heat Equation

31 min
Beginner
Video
Theory

In this video, we derive the heat equation. This partial differential equation (PDE) applies to scenarios such as the transfer of heat in a uniform, homogen...

See More

## Introduction to Partial Differential Equations

52 min
Beginner
Video
Theory

This is the first lesson in a multi-video discussion focused on partial differential equations (PDEs).In this video we introduce PDEs and compare them with o...

See More

## Standard 2nd Order ODEs: Natural Frequency and Damping Ratio

94 min
Beginner
Video
Theory

In this video we discuss writing 2nd order ODEs in standard form xdd(t)+2*zeta*wn*xd(t)+wn^2*x(t)where zeta = damping ratio wn = natural ...

See More

## Numerically Solving Partial Differential Equations

101 min
Beginner
Video
Theory

In this video we show how to numerically solve partial differential equations by numerically approximating partial derivatives using the finite difference me...

See More

## Solving the 1D Heat Equation

47 min
Beginner
Video
Theory

In this video we simplify the general heat equation to look at only a single spatial variable, thereby obtaining the 1D heat equation. We solving the result...

See More

## Standing Waves Demonstration

44 min
Beginner
Video
Theory

In this video we demonstrate standing waves. We show how the system can be excited by oscillating at specific frequencies to generating standing waves. The...

See More