Ordinary Differential Equations
An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. A differential equation is an equation that contains a function with one or more derivatives. But in the case ODE, the word ordinary is used for derivative of the functions for the single independent variable.
In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. The types of DEs are partial differential equation, linear and non-linear differential equations, homogeneous and non-homogeneous differential equation.
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Definition
In mathematics, the term “Ordinary Differential Equations” also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. In other words, the ODE is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives.
y’,y”, ….y^{n} ,…with respect to x.
Order
The order of ordinary differential equations 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 dy/dx or dy/dt and y^{n} can be either d^{n}y/dx^{n} or d^{n}y/dt^{n}.
An n-th order ordinary differential equations 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 ≤ j ≤ n are called the coefficients of the linear equation. The equation is said to be homogeneous if r(x) = 0. If r(x)≠0, it is said to be a non- homogeneous equation. Also, learn the first-order differential equation here.
Types
The ordinary differential equation is further classified into three types. They are:
- Autonomous ODE
- Linear ODE
- Non-linear ODE
Autonomous Ordinary Differential Equations
A differential equation which does not depend on the variable, say x is known as an autonomous differential equation.
Linear Ordinary Differential Equations
If differential equations can be written as the linear combinations of the derivatives of y, then they are called linear ordinary differential equations. These can be further classified into two types:
- Homogeneous linear differential equations
- Non-homogeneous linear differential equations
Non-linear Ordinary Differential Equations
If the differential equations cannot be written in the form of linear combinations of the derivatives of y, then it is known as a non-linear ordinary differential equation.
Applications
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 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
Examples of ODE
Some of the examples of ODEs are as follows;
\(y’=x^2 – 1\\ \frac{d y}{d x}=(x+y)^{6} \\ x y^{\prime}=\sin x \\ \frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+5 y=0, y(0)=0, y^{\prime}(0)=2 \\ y^{\prime \prime}=y^{\prime}+x e^{x}\)Problems and Solutions
The solutions of ordinary differential equations can be found in an easy way with the help of integration. Go through the below example and get the knowledge of how to solve the problem.
Question 1:
Find the solution to the ordinary differential equation y’=2x+1
Solution:
Given, y’=2x+1
Now integrate on both sides,
∫ y’dx = ∫ (2x+1)dx
y = 2x^{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 integrating 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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Frequently Asked Questions – FAQs
What is Ordinary differential equation? Give an example.
What are the types of Ordinary differential equations?
Autonomous Ordinary Differential Equations
Linear Ordinary Differential Equations
Non-linear Ordinary Differential Equations
What is the order of ordinary differential equations?
What is an explicit ordinary differential equation?
F(x, y, y’, …., y^{n-1}) = y^{n}
What is an implicit ordinary differential equation?
F(x, y, y’, y’’, …., y^{n-1}) = 0