Differential Equation
A type of equation in which we have function containing more than one derivatives
For Example:
$$\frac{y+dy}{dx}=6x$$
Fact: It is an equation containing y with derivative
$$\frac{dy}{dx}$$
How to solve Differential Equation
There are many ways to solve differential equation”We will discuss them later
Examples:
-
The Verhulst Equation:
$$\frac{dN}{dt}=rN(\frac{1-N}{k})$$
-
Simple Harmonic Motion:
$$\frac{md2x}{dt2}=-kx$$
Types of differential equation:
- Ordinary Differential Equations(ODEs)
- Partial Differential Equation(PDEs)
Ordinary Differential Equations(ODEs): A type of equation which contain ordinary derivative of a function and it contain one independent variable
Partial Differential Equation(PDEs): A type of equation which contain several independent variables
What is the order and degree of the differential equation?
Order:
It means highest order derivatives involve in differential equation
Degree:
It is power on highest derivative involve in differential equation
Example:
First order: As it contain first derivative:
$$\frac{dy}{dx}+y=4x$$
Order - 1
(as first derivative)
Degree - 1
Example: Second Order: As it contain second derivatives
$$\frac{d^2y}{dx^2}+\frac{dy}{dx}=sinx$$
Order - 2
(as second derivative)
Degree - 1
More Examples: First Order Second Degree Ordinary Differential Equation
$$(\frac{dy}{dx})^2+y=9x$$
Third Order Second Degree Ordinary Differential Equation
$$\frac{d^3y}{dx^3}+(\frac{dy}{dx})^2+y=9x$$
Linear Differential Equation
An equation of the form:
$$\frac{dy}{dx}+P(x)y=Q(x)$$
We will discuss it briefly later.