# Differential Equation And Its Types

Before knowing about Differential Equation and its types, let us know what a differential equation is. An equation with one or more terms that involve derivatives of the dependent variable with respect to an independent variable is known as **Differential Equation.**

An equation with one or more terms that involve derivatives of the dependent variable with respect to an independent variable is known as a differential equation**.**

In simple words, a differential equation consists of derivatives, which could either be ordinary derivatives or partial derivatives.

Example:

\( \frac {d^2y}{dx^2} + \left( \frac {dy}{dx} \right)^2 \)

\( \frac {d^2y}{dt^2} + \frac {dy}{dt} = 5 sin t \)

## Classification of Differential Equation

The below given tree is the general classification of differential equation.

Now let us learn all the types with detailed explanations.

## Differential Equations- Based on Type

There are two types of differential equation:

- Ordinary Differential Equation
- Partial Differential Equation

### Ordinary Differential Equation

Ordinary differential equation which depends on a single independent variable.

Example: \( \frac {dy}{dx} + 5x = 5y \)

__(b) Partial Differential Equation__

It involves partial derivatives.

\( \frac {\partial y}{\partial x} + \frac {\partial y}{\partial t} = x^3 – t^3 \)……….(i)

\( \frac {\partial^2 y}{\partial x^2} – c^2 \frac {\partial^2 y}{\partial t^2} = x0 \) ……………..(ii)

## Differential Equations- Based on Order

The order of the highest differential coefficient (derivative) involved in the differential equation is known as the order of the differential equation.

For Example:- \( \frac {d^3y}{dx^2} + 5 \frac {dy}{dx} + y = \sqrt{x} \)

Here, the order = 3 as the order of the highest derivative involved is 3.

For derivatives the use of single quote notation is preferred which is

\( y’ = \frac {dy}{dx} \).

\( y” = \frac {d^2y}{dx^2} \)

\( y”’ = \frac {d^3y}{dx^3} \)

and so on

For the higher order derivatives it would become cumbersome to use multiple quotes so for these derivatives we prefer using the notation y_{n} for the n^{th} order derivative \( \frac {d^ny}{dx^n} \).

Consider the following examples:-

(i) y” + 5y’ – 6y =\( x^2 \)+ 3x

(ii) x’ = -x + 16

(iii) x”’ + 2x’ = 0

The equation (i) is a second order differential equation as the order of the highest differential co-efficient is 2.

Similarly, the example is a first-order differential equation as the highest derivative is of order 1.

The example is a third-order Differential Equation.

## Differential Equations – Based on Linearity

By linearity, it means that the variable appearing in the equation is raised to the power of one. The graph of linear functions is generally a straight line. For example: (3x + 5) is Linear but (x^{3} + 4x^{2}) is non-linear.

### Linear Differential Equation

If all the dependent variables and its entire derivatives occur linearly in a given equation, then it represents a linear differential equation.

### Non-Linear Differential Equations

Any differential equation with non-linear terms is known as a non-linear differential equation.

Consider the following examples for illustration:

Example 1: \( \frac {dy}{dx} + xy = 5x \) ……… (i)

\( \frac {d^2y}{dx^2} – ln y = 10 \) ………. (ii)

Example 1: \( \frac {dy}{dx} + xy = 5x \)

It is a linear differential equation as \( \frac {dy}{dx} \) and both are linear.

Example 2: \( \frac {d^2 y}{dx^2} – ln \space y =10 \)

In y is not linear. Hence, this equation is non-linear.

## Differential Equations – Based on Homogeneity

Consider the following functions:

\( f_1(x, y) = y^3 + \frac 23 xy^2 \)

\( f_2(x, y) = x^3 ÷ y^2 x \)

\( f_3(x, y) = tan x + sec y \)

If we replace x and y by αx and αy respectively, where α is any non-zero constant, we get;

\( f_1(x,y) = (\alpha y )^3 + \frac 23 (\alpha x) (\alpha y)^2 = \alpha^3 ( y^3 + \frac 23 xy ) = \alpha^3 f_1 (x,y) \)

\( f_2 (x,y) = \frac {(\alpha x)^3}{(\alpha y)^3 (\alpha x)} = \frac {x^3}{xy^2} = \alpha^\circ f_2 (x,y) \)

\( f_3 (x,y) = tan (\alpha x) + sex (\alpha y) \)

We observe that,

\( f_1,f_2 \) can be written in the form \( f(αx, αy) = α^n f(x, y) \) but this is not applicable to \( f_3 (x, y) \) Therefore, if a function satisfies the condition that \( f (αx, xy) = α^n f(x, y) \) for a non-zero constant α, it is known as homogeneous equation of degree n.

The linear differential equation of the form, \( fn (x) y^n + …….+ f_1 (x)y’ + f0(x) y = g(x) \) represents a homogeneous differential equation if the R.H.S is zero i.e.,g(x) = 0 , Elseit represents non-homogeneous differential equation if g(x) \( \ne \) 0.

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