Ordinary Differential Equations

Ordinary Differential Equation

Introduction 

It is the derivative of an unknown function included in an ordinary differential equation. Concerning the independent variable in the ordinary differential equation, there is a derivative of the dependent variable. Homogeneous and non-homogeneous ordinary differential equations can be divided into two groups.

In an ordinary differential equation, just one independent variable can be considered how the dependent variable changes are necessary if you want to know how the independent variable changes about it. A partial or an ordinary derivative is required for an unknown function in a differential equation. One dependent variable is derivatives concerning more than one independent variable in a partial differential equation. We’ll now refer to these as “normal” or “ordinary” differential equations to prevent confusion.

These ordinary differential equations utilise notations for the derivatives.

dy/dx = y’, d2y/dx2 = y”, d3y/dx3 = y”’, dny/dxn = yn.

The following are a few real-world examples of ordinary differential equations.

1. (dy/dx) = sin x
2. (d2y/dx2) + k2y = 0
3. (d2y/dt2) + (d2x/dt2) = x
4. (d3y/dx3) + x(dy/dx) – 4xy = 0
5. (rdr/dθ) + cosθ = 5

Degree and order of Ordinary differential equations

Ordinary differential equations include two fundamental features: the differential equation’s order and degree.

The degree of a differential equation is the integral power of the highest order derivative that arises if the differential equation is expressed in polynomial form. The power of the highest-order derivative is the degree of the differential equation. For the degree of the differential equation to be determined, the index of each derivative must be a positive integer.

Order of ordinary differential equations

The maximum derivative of the dependent variable with respect to the independent variable determines the differential equation’s order. As an example, look at these two differential equations:

dy/dx = ex, 

(d5y/dx5) + x2(d2y/dx2) = 0.

The highest derivatives in these differential equations are first and fifth order.

Ordinary Differential Equations of First Order: The first-order differential equation with a degree of one is known as this equation. Derivatives of all linear equations are in the first order. Only the first derivative, dy/dx, where x and y are the two variables, can be expressed as:

dy/dx = f(x, y) = y’

Ordinary differential equation is second order : In mathematics, a differential equation is a second-order differential equation if it contains the second-order derivative. It has the following graphical representation:

d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.

For example,

(d4y/dx4)3 +4(dy/dx)7+6y=5cos3x

The differential equation has an order of four and a degree of three in this case. There are always positive integers in the order and degree of differential equations. It is also not possible to write a differential equation using the leading term of the highest order polynomial equation, so the degree of a differential equation is not defined.

Differential Equations Types

Homogeneous and non-homogeneous ordinary differential equations are the two broad classifications of ordinary differential equations. In order to better understand these two forms of differential equations, we should take a closer look at each of them.

Homogeneous Differential Equation

A homogeneous differential equation is a differential equation in which the degree of all terms equals one another.

P(x,y)dx + Q(x,y)dy = 0,

Here , P(x,y) and Q(x,y) are homogeneous functions of the same degree, and this can be expressed as. Here are a few instances of Homogeneous Differential Equations:

• y + x(dy/dx) = 0 is a homogeneous differential equation of degree 1
• x⁴ + y⁴(dy/dx) = 0 is a homogeneous differential equation of degree 4
• xy(dy/dx) + y² + 2x = 0 is not a homogeneous differential equation

Non-homogeneous differential equations

A non-homogeneous differential equation is one in which the degrees of all the terms are not the same.

For example, 

xy(dy/dx) + y² + 5x = 0 

The differential equation is not homogenous. An example of a non-homogeneous differential equation is a linear differential equation, which is comparable to a linear equation in many ways.

This type of differential equation has several functions and derivatives in addition to the primary variable. The standard form of a linear differential equation is dy/dx + Py = Q,

The variable y is included, as well as its derivatives. In this differential equation, the numeric constants or functions of x can be P and Q. When solving linear differential equations, the term “linear” is used. We can also write the linear differential equation in x. linear differential equation in terms of x is dx/dy + P1x = Q1

The first-order linear differential equation is a first-order differentiation. The following are some examples of linear differential equations in y are dy/dx + y = Cosx, dy/dx + (-2y)/x = x2.e-x

Furthermore, the linear differential equations in x are dx/dy + x = Siny, dx/dy + x/y = ey. dx/dy + x/(ylogy) = 1/y.

Solution of ordinary differential equation

Ordinary differential equations for a given system y = φ(x)

It is known as the ordinary differential equation solution (integral curve). There are an infinite number of solutions to the ordinary differential equation. In mathematics, the process of finding a solution to a differential equation involves integration, which is why the term “integrating a differential equation” is used. the dependent variable’s expression with respect to the independent variable, which is a solution of an ordinary differential equation

General solution: The solution that includes arbitrary constants is referred to as the “general” solution. Numerous arbitrary constants may be included in the general solution.

Particular solution: The differential equation’s “particular solution” is the solution that does not contain any arbitrary constants and is obtained by substituting values for the arbitrary constants in the general solution.

A first-order differential equation is formed by removing one arbitrary constant, while a second-order differential equation is formed by removing two arbitrary constants. Let’s take a look at how to solve a differential equation using a real-world example. 

(dy/dx) = x2y + y

1. Divide this differential equation by y. (We isolate the variable here)

(1/y)(dy/dx) = (x2 + 1)

y and x are both variables, so we rewrite this as y and x

(dy/y) = (x2 + 1)dx

1. Concerning y and x, we can now integrate LHS and RHS in the same way.

∫(1/y)dy = ∫(x² + 1)dx

1. After the integration, we are left with the following:

log y = (x3/3) + x + c

As a result, since it contains the arbitrary constant C, this is the general solution to the ordinary differential equation. The specific solutions are also available for different values of C.

Conclusion

Calculating the movement or flow of electricity, studying a pendulum’s to and fro motion and checking disease growth in graphic representations are just a few examples of how ordinary differential equations can be put to use in the real world.