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Differential Equation

Posted by Nirazo Malla on 5:47 AM

Any equation, in which derivatives or differentials of one or more dependent variables with one or more independent variables are involved, is called differential equation and is denoted by
$\frac{\text{dy}}{\text{dx}},\frac{\text{dx}}{\text{dy}}$ etc. For eg.
$\text{(i)} \frac{dy}{dx}= \text{sinx}$
$\text{(ii)}\frac{d^2y}{dx^2} + \5 \frac{dy}{dx}+7y = 3$

$\text{In} \frac{dy}{dx}$,
y is dependent variable and x is independent variable.

a) Types of differential equation:-
A differential equation is said to e ordinary diffn if it involves derivatives of dependent variables with respect to single independent variable. For eg.
$\frac{dy}{dx}= \text{sinx}$, $\frac{d^2y}{dx^2} + \5 \frac{dy}{dx}+7y = 3$
b) Partial differential equation:-
A differential equation is said to be partial differential equation if it involves derivative of dependent variables with respect to two or more independent variables. For eg.
$\text{(i)} \frac{\partial^2u }{\partial x^2} + \frac{\partial^2u }{\partial y^2} = \text{0},{u} = \text{f}\left (x,y \right )$
$\text{(ii)} \frac{x\partial z }{\partial x} + \frac{y \partial z }{\partial y} = \text{5z},\text{z}= \text{f}\left (x,y \right )$
Order of differential equation:-
The order of differential equation is the order of highest derivative appearing in it.For eg.
$\text{i)} \frac{\partial ^4y}{\partial x^4}+ \frac{\partial ^2y}{\partial x^2} + \left ( \frac{\partial y}{\partial x} \right )^2 = 5 \to \text{it is the order of 4.}$
$\text{ii)} \frac{\partial ^2y}{\partial x^2}+ \text{3} \frac{\partial y}{\partial x} =2 \to \text{it is the order of 2.}$
$\text{iii)} \sqrt{1 - x^2} \partial x + \sqrt{1 - y^2} \partial y = 0 \to \text{it is the order of 1.}$

Degree of differential equation:-
The degree of differential equation is the degree of highest derivative involving the equation after removing radicals or fractions. For eg.
$\text{i)} 2\frac{\partial ^2y}{\partial x^2} - 5 \frac{\partial y}{\partial x} + 7 = 0 \to \text{it is} \2^{nd} \text {order and} \1^{st} \text{degree}.$
$\text{ii)} \frac{\partial ^2y}{\partial x^2} = \left [ 1 + \left(\frac{\partial y}{\partial x} \right )^2 \right ]^\frac{3}{2}$
becomes

$\left(\frac{\partial ^2y}{\partial x^2} \right )^2 = \left(1 + \left(\frac{\partial y}{\partial x} \right )^2 \right )^3 \to \text {it is} \2^{nd} \text {degree and } \2^{nd} \text {order.}$
Solution of the Differential eqn:-
Solution of the differential eqn is the function relation between the variables involved in differential.
There are two types of solution and they are:-
i) General Solution which gives family of curves and
ii) Particular Solution which gives a particular curves.
Linear differential eqn:-
A differential equation is said to be linear differential equation if dependent variable & derivative involved in it appear only in first degree.

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1 comment:

1. Thanks for giving such useful notes on advance calculus.....

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