Introduction to differential equation

Introduction to differential equation
Written by: Jack Methew

Jack Methew knows that successful students become successful adults. This is her 15th year at Edison Elementary School and her 10th year teaching fourth grade. So far, fourth grade is her favorite grade to teach! Mrs. Carroll was the 2011 Newell Unified School District Teacher of the Year, and received her National Board Certification in 2013. She loves science and majored in biology at Arizona State University, where she also earned her teaching credential and Master of Education degree. Mrs. Carroll is excited to begin the best year ever!

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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:

  1. The Verhulst Equation:

    $$\frac{dN}{dt}=rN(\frac{1-N}{k})$$

  2. Simple Harmonic Motion:

    $$\frac{md2x}{dt2}=-kx$$

Types of differential equation:

  1. Ordinary Differential Equations(ODEs)
  2. 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.