# Introduction to Differential Equations

Table of Contents

A differential equation is an equation involving an unknown function and its derivative [1] , [2] , [3] .

Differential equations are used to model systems and other behaviors in various fields of science, engineering, and mathematics.

## Order of a Differential Equation

The highest derivative included in the differential equation determines the order of the equation.

### Examples

The differential equation
\[ \dfrac{dy}{dx} + 2y = 0 \]
is of the first-order because the highest derivative included is the first derivative \( \dfrac{dy}{dx} \) of \( y \).

The differential equation
\[ \dfrac{d^3y}{dx^3} - 5\dfrac{d^2y}{dx^2} + 6\dfrac{dy}{dx} = 0 \]
is of the third-order because the highest derivative included is the third derivative \( \dfrac{d^3y}{dx^3} \) of \( y \).

The differential equation
\[ \dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + 2y = 0 \]
is of the second-order because the highest derivative included is the second derivative \( \dfrac{d^2y}{dx^2} \) of \( y \).

## Linearity a Differential Equation

Differential equations can be classified based on their linearity.

### Linear Differential Equations

If the highest power of the unknown function and its derivatives is equal to 1 and the function and its derivatives are not multiplied together, then the differential equation is said to be linear. The general form of a linear differential equation can be expressed as:
\[ a_n(x)\dfrac{d^n y}{dx^n} + a_{n-1}(x)\dfrac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x)\dfrac{dy}{dx} + a_0(x)y = g(x) \]
Here, \( a_i(x) \) are functions of \( x \), \( y \) is the unknown function, and \( g(x) \) is a known function of \( x \).
#### Examples

The differential equation
\[ \dfrac{d^2 y}{dx^2} + 2y = 0 \]
is linear because the unknown function \( y \) and its second derivative \( \dfrac{d^2y}{dx^2} \) are not raised to a power greater than 1 or multiplied together.

The differential equation
\[ \dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} + 2 y = 0 \]
is linear because the unknown function \( y \), its second derivative \( \dfrac{d^2y}{dx^2} \) and first drivetive \( \dfrac{dy}{dx} \) are not raised to a power greater than 1 or multiplied together.
### Nonlinear Differential Equations

If the unknown function or at least one of its derivatives are raised to the power greater than 1 or are multiplied together the differential equation is said to be nonlinear. Differential equations involving non linear functions such as square root, logarithmic, exponential, trigonometric, or other non linear functions of the unknown function or its derivatives are also nonlinear.

Nonlinear differential equations may have complex behavior and may be challenging to solve.
#### Examples

The differential equation
\[ \dfrac{dy}{dx} = y^2 + 3 \]
is not linear because because \( y \) appear in raised to the power 2.

The differential equation
\[ \dfrac{d^2y}{dx^2} = e^{y} \]
is not linear because because \( y \) appear in the nonlinear terms \( e^{y} \).

The differential equation
\[ \dfrac{d^2y}{dx^2} = \sin(y) \]
is not linear because \( y \) appears inside a trigonometric function \( \sin(y) \) which is nonlinear function.

### More References and links

1 - University Calculus - Early Transcendental - Joel Hass, Maurice D. Weir, George B. Thomas, Jr., Christopher Heil - ISBN-13 : 978-0134995540

2 - Calculus - Gilbert Strang - MIT - ISBN-13 : 978-0961408824

3 - Calculus - Early Transcendental - James Stewart - ISBN-13: 978-0-495-01166-8

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