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DifferentialEquations Definition


DifferentialEquations Definition:

           An equation involving differentials or one dependent variable and its derivatives with respect to one or more independent variables is called differential equation.

Ordinary Differential Equation:

              A differential equation is said to be ordinary, if the derivatives in the equation have reference to only a single independent variable.

Example:

   1. \(\left ( \frac{dy}{dx} \right )^{3}-4\left ( \frac{dy}{dx} \right )^{2}+7y=\cos (x)\)

2. \(\left ( \frac{d^{2}y}{dx^{2}} \right )+4x\left ( \frac{dy}{dx} \right )^{2}+7y=\ln (x)\)

The general form of an ordinary differential equation is \(F\left ( x,y,{y}',{y}''\cdots y^{(n)} \right )= 0\)


Partial Differential Equation:

A differential equation is said to be partial, if the derivatives in the equation have reference to two or more independent variables.

Examples:

1) (y+z)\(\frac{\partial z}{\partial x}\) + (z + x) \(\frac{\partial z}{\partial x}\) = x + y

Here in this x and y are two independent variables.

2)\(\left ( \frac{\partial z}{\partial x} \right )^{2}\) + \(\left ( \frac{\partial z}{\partial x} \right )^{2}\) = 4z

3) \(4 \frac{\partial^{2}U}{\partial x^{2}} + 5 \frac{\partial^2 U }{\partial x \partial y} + 3 \frac{\partial^{2}U}{\partial y^{2}}\) = x + 2y

Solutions of a Differential Equation :

Definition:

    A relation between the dependent and independent variables when substituted in the differential equation reduces it to an identity, is called a solution or integral or primitive of the differential equations.

Note:

  A solution of differential equation does not involve the derivatives of the dependent variables with respect to the independent variable.

These are basic definitions of Differential Equations .

Further it has many methods to solve Differential Equations.

Basic methods are Variable separable, Exact Differential Equtions, Non-Exact Differential Equations, Linear Differential Equations, Bernoulli Differential Equations, Cauchy's Differential Equations etc .

Let us discuss each topic in detail with sample questions and their solutions .

DifferentialEquations Definition

Differential Equations


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