What are Ordinary Differential Equations (ODE)?

ODEs

A differential equation is any equation containing either ordinary or partial derivatives. In applied mathematics, an Ordinary Differential Equation (ODE) is a differential equation with functions of one independent variable and its derivatives. In other words, ODE is represented as a relation with one independent variable and one or more real dependent variables.

Given a function $F$ of $x, y,$ and derivatives of $y$ the general form is as follows:

Note: Ordinary differential equations are employed in contrast to partial differential equations, which may refer to more than one independent variable.

Order

The order of a differential equation refers to the order of the highest derivative in the equation.

Example

Consider the following ODE:

Since only one derivative is involved so the order is $1$. The polynomial equations with order $1$ is known as $1^{st}$order ODEs.

Let's consider another example of ODE containing order $= 2$.

Here, we'll only focus on the term with the highest derivative. The polynomial equations of order $= 2$ is called $2^{nd}$ order ODEs.

Degree

The power of the highest order derivative in the specified equation represents the degree of the ODE.

Note: For the degree to be defined, the ODE must be a polynomial equation with derivatives.

Example

Consider the following ODE:

The highest derivative in this equation is of order 2, and the exponent raised to the highest derivative is 2.

Classification

ODEs are classified into multiple types. Some of the widely known are defined below:

Autonomous ODE

An autonomous ODE is stated as follows:

It's independent of $x$ and separable, allowing a solution to be obtained by integrating.

Linear ODE

Ordinary linear differential equations are expressed as linear combinations of the derivatives of $y$ as follows:

Non-linear ODE

Non-linear ODE is the one in which differential equations cannot be expressed as linear combinations of the derivatives of $y$.

Example

Bellman's equation is one of the examples of non-linear ODE:

Homogeneous ODE

Homogeneous ODE contains only a derivative of the dependent variable and terms involving the dependent variable, which is set to $0$.

For example, consider the following equation:

Non-homogeneous ODE

Non-homogeneous differential equations are differential equations that do not satisfy the requirements for homogeneous equations. We discovered that homogeneous equations have zero on the right side of the equation, whereas non-homogeneous ODEs are those that have a function on the right side of their equation

Types of solutions

The ordinary differential equation has an infinite number of solutions. An ODE solution is an expression of the dependent variable with reference to the independent variable that satisfies the equation.

General solution

The general solution is the one that involves arbitrary constants. It contains many arbitrary constants.

Particular solution

The particular solution of ODE is the solution that is free of arbitrary constants and is derived by substituting values for the arbitrary constants in the general solution.

Note: Eliminating one arbitrary constant results in a $1^{st}$ order differential equation, while eliminating two arbitrary constants results in $2^{nd}$ order differential equation, and so on.

Applications

Ordinary differential equations (ODEs) appear in a wide range of contexts in mathematics, as well as in social and natural sciences. Differentials and derivatives are used in the mathematical representation of change. Some of the practical applications are as follows:

• It models the growth of diseases.

• It's used in the second law of motion and by Newton, the law of cooling.

• It describes the pendulum and waves motion.

• It calculates the flow of electricity.