Overview on types of Ordinary Differential Equations

The derivative of an unknown function is contained in an ordinary differential equation. An ordinary differential equation has variables and a derivative of the dependent variable with respect to the independent variable. The homogeneous differential equation and the non-homogeneous differential equation are the two forms of ordinary differential equations.

A differential equation with ordinary derivatives is known as an ordinary differential equation (ODE) (and NOT the partial derivatives). An equation with variables and a derivative of the dependent variable with reference to the independent variable is known as a differential equation. A differential equation contains at least one unknown function’s derivative, either an ordinary or partial derivative. Ordinary derivatives are found in ordinary differential equations. Ordinary differential equations are referred to as just differential equations in this context.

In these ordinary differential equations, the derivatives are written as dy/dx = y’, d2y/dx2 = y”, d3y/dx3 = y”‘, and dny/dxn = yn. Here are a few examples of ordinary differential equations.

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

Order and degree of ordinary differential equation:

The order and degree of the differential equation are two fundamental characteristics of ordinary differential equations.

Order of ordinary differential equations:

The maximum derivative of the dependent variable with respect to the independent variable determines the order of a differential equation. Consider the differential equations dy/dx = ex, (d4y/dx4) + y = 0, (d3y/dx3) + x2(d2y/dx2) = 0, (d3y/dx3) + x2(d2y/dx2) = 0. The largest derivatives in these differential equations are of first, fourth, and third order, respectively, and their orders are 1, 4, and 3.

First-order differential equation:

It is a first-order differential equation with a degree of one.All derivatives of linear equations are in the first order. It simply has the first derivative, such as dy/dx, where x and y are the two variables, and is written: dy/dx = f(x, y) = y’

Second-order differential equation:

The second-order differential equation is an equation that includes a second-order derivative. d/dx(dy/dx) = d2y/dx2 = f”(x) = y” is how it’s written.

Degree of ordinary differential equations:

The degree of a differential equation is the integral power of the highest order derivative that arises when a differential equation can be represented in polynomial form. The power of the highest order derivative in the differential equation determines the degree of the equation. We require a positive integer as the index of each derivative to obtain the degree of the differential equation. Example:

        (d4y/dx4)3 + 4(dy/dx)7 + 6y = 5Cos3x

The differential equation’s order is 4 and the degree is 3. A differential equation’s order and degree are always positive integers. Furthermore, if a differential equation cannot be expressed in terms of a polynomial equation with the highest order derivative as the leading term, the differential equation’s degree is undefined.

Types of ordinary differential equations:

There are three different types of ordinary differential equations. They are:

• ODE with autonomy
• ODE linear
• ODE with non-linearity

Autonomous ordinary differential equations:

An autonomous differential equation is a differential equation that does not depend on a variable, such as x.

Linear ordinary differential equations:

Linear ordinary differential equations are differential equations that can be represented as linear combinations of the derivatives of y. Now it is further classified into two categories:

• Linear differential equations with homogeneous coefficients
• Linear differential equations with non-homogeneous solutions. 

Non-linear ordinary differential equations:

A non-linear ordinary differential equation is one in which the differential equations cannot be stated as linear combinations of the derivatives of y.

Conclusion:

A differential equation having one or more functions of one independent variable and their derivatives is known as an ordinary differential equation (ODE) in mathematics. The derivative of an unknown function is contained in an ordinary differential equation.

The order and degree of the differential equation are two fundamental characteristics of ordinary differential equations.

The maximum derivative of the dependent variable with respect to the independent variable determines the order of a differential equation.bIt is a first-order differential equation with a degree of one. The second-order differential equation is an equation that includes a second-order derivative.

The degree of a differential equation is the integral power of the highest order derivative that arises when a differential equation can be represented in polynomial form. The power of the highest order derivative in the differential equation determines the degree of the equation. An autonomous differential equation is a differential equation that does not depend on a variable, such as x. A non-linear ordinary differential equation is one in which the differential equations cannot be stated as linear combinations of the derivatives of y.