Ordinary Differential Equations (Types, Solutions & Examples) (2024)

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.

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”, ….yⁿ,…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’,….,yⁿ) = 0

Note that, y’ can be either dy/dx or dy/dt and yⁿcan be either dⁿy/dxⁿor dⁿy/dtⁿ.

An n-th order ordinary differential equations is linear if it can be written in the form;

a₀(x)yⁿ + a₁(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 hom*ogeneous if r(x) = 0. If r(x)≠0, it is said to be a non- hom*ogeneous 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:

• hom*ogeneous linear differential equations
• Non-hom*ogeneous 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;

\(\begin{array}{l}y’=x^2 – 1\\ \frac{d y}{d x}=(x+y)^{6} \\ x y^{\prime}=\sin x \\ \frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+5 y=0, y(0)=0, y^{\prime}(0)=2 \\ y^{\prime \prime}=y^{\prime}+x e^{x}\end{array} \)

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 = 2x²/2 + x + C

y =x²+ x + C

Where C is an arbitrary constant.

Question 2:

Solve y⁴y’+ y’+ x² + 1 = 0

Solution:

Take, y’ as common,

y'(y⁴+1)=-x²-1

Now integrating on both sides, we get

\(\begin{array}{l}\frac{y^{5}}{5}+y=-\frac{x^{3}}{3}-x+C\end{array} \)

Where C is an arbitrary constant.

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