Introduction to Differential Equations

In this post I am going to write about the fundamentals of Differential equations. But before we go any further let's remind the definition of equation.
An equation is a mathematical statement that two things are equal.
Ex: 15 = 9 + 6
Both sides of the sign equal have the same value.
An equation can also involve an unknown quantity
Ex: x + 5 = 9
Solving an equation is to find the value of the unknown. This unknown can have one or more values. It can also have an infinity of values.

Differential equation. Definition

A differential equation is an equation that is differential. Differential is an adjective that derives from the word "derivative". A differential equation is an equation that involves a function and its derivative. Solving a differential equation is to find the value of the unknown of this equation. The unknown is a function. Finding the solution of a differential equation is finding the value of the unknown function.

The solution of a differential equation can be one function, a set of functions or a class of functions.
Example: y + dy/dx = 7
y is the unknown function to find. dy/dx is its derivative.
 This equation can also be written as y + y' = 7 or f(x) + f'(x) = 7.

Ordinary differential equation

An ordinary differential equation is an equation where the unknown function has one independent variable.

Examples:

Partial differential equation

A partial differential equation is an equation where the unknown function has more than one independent variable.

Examples:

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Order

The order of a differential equation is the highest derivative.

Examples:

dy/dx + y = 2x + 1 is a first order differential equation

dy/dx + d²y/dx² is a second order differential equation.

Linear differential equations

An ordinary differential equation has the form:

In a linear differential equation, there are no products of the function and its derivatives. The function and its derivatives have only the power of 1.

Examples:

2 y''' + 5y'' + 3y' + 6y = sint

cosydy/dx = y²dy/dx + y³ is non linear. The function has the power 2.

Interested in reviewing in a simple manner the notions of limit, continuity, derivatives and integral visit
Center for Integral Development