Question

Mathematics

Explain Differential equation​

Answer 1

Answer:

*A real valued function y=f(x) is said to be differentiable or derivable at a point of its derivative dy/DX or f'(x) exist at that point.*

hope this will help you

Answer 2

$\large\underline{\sf{Solution-}}$

Differential Equation :-

The equation consist of derivative of the dependent variable with respect to independent variable or variables is called Differential equation.

For example :-

$\red{ \boxed{ \rm{ \: y\frac{dy}{dx} + {x}^{2} = 0}}}$

$\red{ \boxed{ \rm{ \: \frac{ {d}^{3} y}{ {dx}^{3} } + { \bigg(\dfrac{dy}{dx} \bigg)}^{2} = 0}}}$

Order of Differential equation

The order of Differential equation is defined as the order of the highest order derivative exist in the given Differential equation.

For example,

The order of

$\red{ \boxed{ \rm{ \: y\frac{dy}{dx} + {x}^{2} = 0 \: is \: 1}}}$

$\red{ \boxed{ \rm{ \: \frac{ {d}^{3} y}{ {dx}^{3} } + { \bigg(\dfrac{dy}{dx} \bigg)}^{2} = 0 \: is \: 3}}}$

Degree of Differential equation

The degree of Differential equation is defined as the degree of higher order derivative exist in the given Differential equation when all the other Differential coefficients are free from radicals.

Degree is defined when Differential equation is a polynomial in Differential coefficient otherwise degree is not defined.

For example

The degree of

$\red{ \boxed{ \rm{ \: y\frac{dy}{dx} + {x}^{2} = 0 \: is \: 1}}}$

$\red{ \boxed{ \rm{ \: \frac{ {d}^{3} y}{ {dx}^{3} } + { \bigg(\dfrac{dy}{dx} \bigg)}^{2} = 0 \: is \: 1}}}$

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} + cos \bigg(\dfrac{dy}{dx} \bigg) = 0 \: is \: not \: defined}}}$

Solution of differential equation

General Solution : -

The solution which contains arbitrary constants is called general solution. The number of arbitrary constants is always equals to order of Differential equation.

Particular solution :-

The solution which don't have any arbitrary constants is called Particular solution of differential equation.

Method of solving first order and one degree differential equations :-

1. Method of Variable separations.

They are of the form :-

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} = f(x)}}}$

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} = f(y)}}}$

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} = f(x) \: f(y)}}}$

2. Method of solving Homogeneous differential equation of degree 0.

They are of the form :-

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} = f \bigg(\dfrac{y}{x} \bigg)}}} \: \: or \: \: \: \red{ \boxed{ \rm{ \: \frac{dx}{dy} = g \bigg(\dfrac{x}{y} \bigg)}}}$

3. Method of solving Linear Differential equation.

They are of the form :-

$\red{ \boxed{ \rm{ \: \frac{dy}{dx} + py = q, \: where \: p,q \: \in \: f(x)}}}$

$\red{ \boxed{ \rm{ \: \frac{dx}{dy} + px = q, \: where \: p,q \: \in \: f(y)}}}$