Ordinary Differential Equations

Ordinary differential equations appear in many different contexts, including mechanics, astronomy, geometry, and population modelling. Many distinguished mathematicians have contributed to this field, including Newton, Clairaut, d'Alembert, Euler and many more.

An ordinary differential equation (also known as ODE), in the subject of Mathematics, is a type of equation that consists of one or more functions of an individual independent variable along with their derivatives.

Definition of Ordinary Differential Equation

An ordinary differential equation (ODE) is a differential equation in mathematics that has one or more functions of one independent variable and its derivatives. The word ordinary differential equation is used in contrast to the term partial differential equation, which can refer to more than one independent variable.

Ordinary Differential Equation In Different Ways

As we know, a differential equation includes a function with one or more derivatives. However, in the case of an ODE, the term ordinary is used for the derivative of the functions for the individual independent variable.

In the case of other types of differential equations, it is probable to have derivatives for functions of more than a single variable. There are three kinds of Differential Equations.

• Ordinary Differential Equations.

• Homogeneous Differential Equations.

• Non-homogeneous Differential Equations.

In other words, in mathematics, the phrase "Ordinary Differential Equations," which is commonly known as ODE, is a kind of equation that contains only a single independent variable and a single or more than one of its derivatives with respect to the concerned variable.

To put it plainly, the Ordinary Differential Equation is represented as the relation having one independent variable (such as) 'x,' the original dependent variable 'y,' with some of its derivatives.

Types of Ordinary Differential Equations

The Ordinary Differential Equations can be of various types, and their application is in different mathematics areas. We can primarily classify ODE or Ordinary Differential Equations into three types. They are stated below:

• Autonomous Ordinary Differential Equation

• Linear Ordinary Differential Equations

• Non-linear Ordinary Differential Equations

What are Autonomous (ODE) Ordinary Differential Equations?

A differential equation that is not dependent on the variable, such as x, is known as an autonomous differential equation.

Example: $\dfrac{dy}{dv}=by+9$

What are Linear (ODE) Ordinary Differential Equations?

Those differential equations that can be written as the linear combinations of the derivatives of, say y, then they are known as linear ordinary differential equations. This type of equation can be further grouped into two types as mentioned below:

• Non-homogeneous linear differential equations

• Homogeneous linear differential equations

An example of Linear Ordinary Differential Equation is $\dfrac{dy}{dx}+2y=sin(x)$

What are Non-linear (ODE) Ordinary Differential Equations?

Those differential equations that cannot be written in the style of linear combinations of the derivatives of, say y, then they are known as a non-linear ordinary differential equation.

An example of Linear Ordinary Differential Equation is y′′= f(x, y)

Examples of Ordinary Differential Equations

While solving any Ordinary Differential Equations, an individual must be aware of the two types of equations they may encounter in mathematics. The two kinds of Ordinary Differential Equations are first-order and second-order differential equations. An individual must perform at least one integration.

The difference between a first and second-order differential equation is in the number of constants one gets upon solving the Differential Equation. One constant indicates it is a first-order, and obtaining two constants means the Equation is a second-order, and so on. An individual should assume here that there are no boundaries or initial values involved.

• Example of first-order equation: $y+\dfrac{py}{px}=5x$

• Example of first-order equation: $xy+\dfrac{p^{2}y}{px^{2}}=sin(x)$

Difference Between ODE and PDE

The differences between ODE and PDE are:

• ODEs involve derivatives including only one variable, whereas PDEs involve derivatives including multiple variables. Therefore, all ODEs can be termed as PDEs.

• PDEs are usually more challenging to understand when solutions are concerned than ODEs.

• Every central theorem about ODEs does not apply to PDEs.

• For an ODE, we can frequently view the single independent variable as a time variable so that ODEs command the flow of an object in time.

• The concept of ODEs overseeing "motion" allows us to use many mathematical results in physics, for example, in empirical behavior regarding Newton's law. It will enable us to understand the solutions much more accurately.

• ODE stands for Original Differential Equation, and PDE stands for Partial Differential Equation.

Solved Examples

Ordinary Differential Equations: Problems and Solutions

Some problems of Ordinary Differential Equations and their solutions are mentioned below in detail.

Example:1

Find the solution to the given ordinary differential equation y’=2p+1

Solution

Given to us, y’=2p+1

Now integrate on both sides,

∫ y’dp = ∫ (2p+1)dp

$y=\dfrac{2p^{2}}{2}+p+C$

y =p2 + p + C

Where C is considered to be an arbitrary constant.

Example: 2

Solve p4p’+ p’+ x2 + 1 = 0

Solution:

Take, p’ as common,

p'(p4+1)=-x2-1

Now, by applying integration on both sides of the equation, we get

$\dfrac{p^{5}}{5}+p=(-\dfrac{x^{3}}{3}-x+C)$

Where C is considered to be an arbitrary constant.

Practice Question MCQs

1. We analyse the system's initial conditions while solving the ordinary differential equation with the unilateral Laplace transform.

1. True

2. False

Answer: A) True

2. If there is no function on the right-hand side of the equation in standard form and the initial conditions are zero, it is feasible to solve an Ordinary Differential Equation using the unilateral Laplace Transform.

1. True

2. False

Answer: B)False

Conclusion

Differential equations play a critical role in the mathematical modelling of physical systems. Many fundamental physics and chemistry rules may be expressed as differential equations. Differential equations are used to predict the behaviour of complex systems in biology and economics.