Order of Differential equations

The highest derivative in a differential equation is called the order of a differential equation.

Introduction

The differential equations mainly contain variables, constants and derivatives. The derivative of a function can be either at least once or none in every term of a differential equation. This factor plays a vital role in solving the differential equations in calculus. Therefore, It is very important to study the concept of the order of differential equations.

$\dfrac{dy}{dx}+3y = 7$

In this example differential equation, the variable $y$ represents a function in $x$. Actually, the function $y$ is differentiated only once and there is no derivatives in the remaining terms of the equation. Hence, the order of this differential equation is one.

Examples

Now, let’s learn the concept of order of differential equations much clear from some more understandable examples.

$(1)\,\,\,$ $x^2\dfrac{d^2y}{dx^2}+4x\dfrac{dy}{dx}+2y$ $\,=\,$ $\log{x}$

In the left hand side of the equation, there are two derivatives as coefficients in the terms but there is no derivatives in the right hand side of the equation. So, we have to identity the highest derivative. The variable $y$ that represents a function in $x$ is differentiated two times highly. Hence, the order of this equation is $2$. So, it is a second order differential equation.

$(2)\,\,\,$ $4\dfrac{du}{dt}$ $\,=\,$ $\Big(\dfrac{d^3u}{dt^3}\Big)^{123}+7$

In this equation, the variable $u$ that represents a function in $t$ is differentiated three time highly but don’t get confused with the exponent $123$ because the high power does not consider in determining the order of the differential equation. The order of this differential equation is $3$. Hence, it is called the third order differential equation.

$(3)\,\,\,$ $\Bigg[\dfrac{d^6z}{dy^2}+x\dfrac{dz}{dy}-2z\Bigg]^{-3}$ $\,=\,$ $z-\sin{z}+8$

In this example, we should not think about the power $-3$ but we have to identify the highest derivative. In this case, the highest derivative is $6$. Hence, the order of this differential equation is $6$ and it is called as sixth order differential equation.

In this way, we determine the order of any differential equation in the differential calculus.