# Ordinary Differential Equations

## Definition: Ordinary Differential Equation

An Ordinary Differential Equations, commonly called as ODEs is a relation involving one real variable x , the real dependent variable y, and some of its derivatives

${y}’,{y}”,…,y^{n},..$ with respect to x

The order of an ODE is defined to be the order of the highest derivative that occurs in the equation. The general form of n-th order ODE is given as

$F(x,y,{y}’,…y^{n})=0$

Note that ,${y}'$ can be either $\frac{dy}{dx}$ or $\frac{dy}{dt}$ and $y^{n}$ can be either $\frac{d^{n}y}{dx^{n}}$ or $\frac{d^{n}y}{dt^{n}}$

An n-th order ODE is linear if it can be written in the form

$a_{0}(x)y^{n}+a_{1}(x)y^{n-1}+….+a_{n}(x)y=r(x)$

The function $a_{j}(x),0\leq j\leq n$ are called the coefficients of the linear equation. The equation is said to be homogeneous if $r(x)\equiv 0$.If $r(x)\not\equiv 0$ , it is said to be a non- homogeneous equation.

## Ordinary Differential Equations Application

ODEs has remarkable applications and it has the ability to predict the world around us. It is used in a variety of disciplines like biology, economics, physics, chemistry and engineering. It helps to predict the exponential growth and decay, population and the species growth. Some of the uses of ODEs are:

• Modelling the growth of diseases
• Describes the movement of electricity
• Describes the motion of the pendulum, waves
• Used in Newton’s second law of motion and Law of cooling.

## Ordinary Differential Equations Examples

Some of the examples of ODEs are as follows

• ${y}’=x^{2}-1$
• $\frac{dy}{dx}=(x+y)^{6}$
• $x{y}’=\sin x$
• $\frac{d^{2}y}{dt^{2}}+2\frac{dy}{dt}+5y=0,y(0)=0,{y}'(0)=2$, (Initial Value Problem)
• ${y}”={y}’+xe^{x}$

## Question 1:

Find the solution to the ordinary differential equation ${y}’=2x+1$

### Solution:

Given, ${y}’=2x+1$

Now integrate on both sides, $\int {y}’dx=\int (2x+1)dx$

Which gives, $y=2\frac{x^{2}}{2}+x+c$

$y=x^{2}+x+c$

Where c is an arbitrary constant.

### Question 2:

Solve $y^{4}{y}’+{y}’+x^{2}+1=0$

### Solution:

Take, ${y}'$ as common,

${y}'(y^{4}+1)=-x^{2}-1$

Now integrate on both sides ,we get

$\frac{y^{5}}{5}+y=-\frac{x^{3}}{3}-x+c$

Where c is an arbitrary constant.

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#### Practise This Question

The product of two improper fractions is greater than either of the fractions.