 # 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 involves derivatives of the dependent variable with respect to an independent variable is known as

Differential Equation

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

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

Example:

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

$\frac {d^2y}{dt^2} + \frac {dy}{dt} = 5 sin t$ (A) Differential Equation and its types- Based on Type:-

(a)  Ordinary Differential Equation:-

It’s a differential equationwhich depends ona single independent variable.

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

In the above mentioned equation,is a function of  only so it’s an ordinary differential equation.

(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)

(B)  Differential Equations and its types- 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 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 yn  for the nth 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 highest differential co-efficient is 2.

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

The exampleis a third order Differential Equation

(C) Differential Equation and its types- Based on Linearity:-

By linearity, it means that 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 (x3 + 4x2) 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.

(b) Non-Linear Differential Equation:-

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

Consider the following examples for illustration:

eg: – $\frac {dy}{dx} + xy = 5x$    ……… (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.

(D) Differential Equation and its types- 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  and byαx  and α y 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.