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.

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

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

__(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

\( y’ = \frac {dy}{dx} \)

\( y” = \frac {d^2y}{dx^2} \)

\( y”’ = \frac {d^3y}{dx^3} \)

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 =\( x^2 \)

(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 (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.

(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: – \( \frac {dy}{dx} + xy = 5x \)

\( \frac {d^2y}{dx^2} – ln y = 10 \)

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

It is a linear differential equation as \( \frac {dy}{dx} \)

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

\( f_1,f_2 \)

The linear differential equation of the form,\( fn (x) y^n + …….+ f_1 (x)y’ + f0(x) y = g(x) \)

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