What are the differential equations? Types of Differential Equations

Historical Background

Differential Equations have already been proved a significant part of Applied and Pure Mathematics since their introduction with the invention of calculus by Newton and Leibniz in the mid-seventeenth century. Differential Equations played a pivotal role in many disciplines like Physics, Biology, Engineering, and Economics.

Applications of Differential Equations

  • Exponential Growth and Decay
  • Population Growth
  • Motion of objects declining under gravity with air resistance and motion of objects hanging from a spring
  • Newton’s Law of Cooling
  • Particle moving on a curve
  • Electrical Circuits
  • Computer Science

What are Differential Equations?

An equation that includes at least one derivative of a function is called a differential equation. Few examples of differential equations are given below.

\displaystyle \frac{{dy}}{{dx}}={{x}^{2}}-10

\displaystyle \frac{{dy}}{{dx}}=2{{x}^{2}}

\displaystyle \frac{{dy}}{{dx}}=x\sin x

\displaystyle \frac{{dy}}{{dx}}+y=16x

Before proceeding further, it is essential to know about basic terms like order and degree of a differential equation which can be defined as,

i. Order – It is the highest derivative of a differential equation, for example,

\displaystyle \frac{{dy}}{{dx}}=12{{x}^{2}}y

The above differential equation has only first derivative i.e. \displaystyle \frac{{dy}}{{dx}}, therefore, it is called a first order differential equation.

Let’s check another differential equation,

\displaystyle \frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}=xy

In this example, differential equation has a second derivative i.e. \displaystyle \frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}, therefore, it is called a second order differential equation.

ii. Degree – It is the exponent of the highest derivative of a differential equation, for example,

\displaystyle \frac{{dx}}{{dt}}=12x-4

In this example, the highest derivative is one and the exponent is also one, therefore, it is called a first order and first degree differential equation. Similarly,

\displaystyle {{\left( {\frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}} \right)}^{3}}=xy

Here, the highest derivative is 2 while the exponent is 3, therefore, it is called a 2nd order and 3rd degree ordinary differential equation.

Types of Differential Equations:

  1. Ordinary Differential Equation
  2. Partial Differential Equation
  3. Linear Differential Equation
  4. Non-linear Differential Equation
  5. Homogeneous Differential Equation
  6. Non-homogeneous Differential Equation

A detail description of each type of differential equation is given below: –

1 – Ordinary Differential Equation

It is a differential equation that involves one or more ordinary derivatives but without having partial derivatives. Ordinary differential equation is different from partial differential equation where some independent variables relating partial derivatives whereas, differential equation has only one independent variable like y. Newton’s 2nd law of motion is the simple example of ordinary differential equation.

\displaystyle m\frac{{{{d}^{2}}x}}{{d{{t}^{2}}}}=f(x)

Another example of ordinary differential equation is;

\displaystyle \frac{{dx}}{{dt}}=12x-4

2 – Partial Differential Equation

Partial differential equation is a differential equation that involves partial derivatives. It has two or more independent variables. For example,

\displaystyle \frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}}+4xy\frac{{{{\partial }^{2}}u}}{{\partial {{y}^{2}}}}+u=2{{\left( {\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}}} \right)}^{2}}+4\frac{{{{\partial }^{3}}u}}{{\partial x\partial {{y}^{2}}}}=10x

3 – Linear Differential Equation

It is first degree with respect to the dependent variable(s) and its derivatives that can be expressed in the form

\displaystyle \frac{{dy}}{{dx}}+p(x)y=q(x) where, p and q can be constants or functions of independent variable x.

4 – Non-Linear Differential Equation

It is second degree or higher with respect to dependent variables and its derivatives. For example,

\displaystyle \frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}-{{\left( {\frac{{dy}}{{dx}}} \right)}^{2}}+12y=\cos x

5 – Homogeneous Differential Equation

It is first order differential equation which can be in written as

\displaystyle {{y}^{{''}}}+f(x){{y}^{'}}+g(x)y=0

\displaystyle \frac{{dy}}{{dx}}=\frac{{f(x,y)}}{{g(x,y)}}

Where, f and g are homogeneous function of similar degree of x and y.  

For examples,

\displaystyle \frac{{dy}}{{dx}}=\frac{{{{x}^{2}}-{{y}^{2}}}}{{x+y}} \displaystyle \frac{{dy}}{{dx}}=\frac{{4{{x}^{2}}}}{{x-y}}

6 – Non-homogeneous Differential Equation

It is a differential equation whose right-hand side is not equal to zero. A 2nd order non-homogeneous equation can be written in this form

\displaystyle {{y}^{{''}}}+f(x){{y}^{'}}+g(x)y=r(x)

For examples,

\displaystyle 4{{y}^{{''}}}+2y=5

\displaystyle 4{{y}^{'}}+25=10

See also : Types of Equations

Types of Differential Equations
Types of Differential Equations