Differential equations and their types‌

Linear equations contain one or more variables of degree one. Quadratic equations contain one or more unknown variables in degree two. Differential equations are the most advanced equations in Mathematics and contain one or more unknown variables to any degree. They contain derivatives of order 1, or at max, derivatives of order 2. They are represented as-

f(x) = dy/dx

In this, y is a dependent variable, and x is the independent variable. dy/dx is known as a derivative. Derivatives are defined as functions that change their rate with respect to another function. Differential equations are used in various fields such as engineering, biology, physics, etc. The solutions of differential equations hold more value than the differential equations themselves.

Order of differential equations

The order of the differential equation is the order of the highest derivative present in the differential equation. For example,

1) dy/dx = 5x – 6. The order of this differential equation is 1.

2) (d³y/dx³) + 9 (dy/dx) – y = 6. The order of this differential equation is 3.

Degree of differential equations

The degree of a differential equation is defined as the degree of the highest order derivative in a differential equation. For example,

1). (dy/dx)⁵ – (d³y/dx³) – 6 = y. The degree of this differential equation is 1, and not 5. We need to find the degree of the highest order derivative to find the degree of the whole differential equation.

2). (d⁹y/dx⁹)⁴ – (d²y/dx²)⁹ + (d⁶y/dx⁶)³ = 0. The degree of this differential equation is 4 and the order is 9.

Types of differential equations

1. Ordinary differential equation – An ordinary differential equation has a function and its derivative. It contains only one independent variable but can contain one or more derivatives. The following methods can solve these differential equations –

a. Separating variable method

b. Integrating factor method

2. Homogeneous differential equations – A differential equation with degree the same in all its terms is known as a homogeneous differential equation. These equations are represented in the form P (x,y) dx + Q (x,y) by = 0, where P (x,y) and Q (x,y) are homogenous functions of the same degree.

3. Nonhomogeneous differential equations – Differential equations with different degrees in their terms is known as a nonhomogeneous differential equation. For example, y (d³y/dx³) + y⁵ = 0 is a nonhomogeneous differential equation as the degree is different in the derivative and y.

4. Partial differential equation – An equation involving one or more partial derivatives of an independent variable is known as a partial differential equation.

5. Linear differential equation – The differential equations with linear order of their derivative are known as linear differential equations. For example, (dy/dx)³ + y = 0. This is a linear differential equation as it contains linear order of its derivative.

6. Non-Linear differential equation – These contain higher orders of derivatives with more than degree one.