How to Find Solutions to System of Differential Equations

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 2^nd order and 3^rd degree ordinary differential equation.

Types of Differential Equations:

Ordinary Differential Equation
Partial Differential Equation
Linear Differential Equation
Non-linear Differential Equation
Homogeneous Differential Equation
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 2^nd 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,

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