Differential Equations

In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.

One of the easiest ways to solve the differential equation is by using explicit formulas. In this article, let us discuss the definition, types, methods to solve the differential equation, order and degree of the differential equation, ordinary differential equations with real-word examples and a solved problem.

Differential Equation Definition

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable)

dy/dx = f(x)

Here “x” is an independent variable and “y” is a dependent variable

For example, dy/dx = 5x

A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity. There are a lot of differential equations formulas to find the solution of the derivatives.

Order of Differential Equation

The order of the differential equation is the order of the highest order derivative present in the equation. Here some examples for different orders of the differential equation are given.

dy/dx = 3x + 2 , The order of the equation is 1

(d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2

(dy/dt)+y = kt. The order is 1

First Order Differential Equation

You can see in the first example, it is a first-order differential equation which has degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as:

dy/dx = f(x, y) = y’

Second-Order Differential Equation

The equation which includes the second-order derivative is the second-order differential equation. It is represented as;

d/dx(dy/dx) = d2y/dx2 = f”(x) = y”