# Order and Linearity of Differential Equations

A tutorial on how to determine the order and linearity of a differential equations.

## Order of a Differential Equation

The order of a differential equation is the order of the highest derivative included in the equation.__Example 1:__State the order of the following differential equations

\dfrac{dy}{dx} + y^2 x = 2x \\\\
\dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} + y = 0 \\\\
10 y" - y = e^x \\\\
\dfrac{d^3}{dx^3} - x\dfrac{dy}{dx} +(1-x)y = \sin y

__Solution to Example 1__

1. The highest derivative is dy/dx, the first derivative of y. The order is therefore 1.

2. The highest derivative is d

^{2}y / dx

^{2}, a second derivative. The order is therefore 2.

3. The highest derivative is the second derivative y". The order is 2.

4. The highest derivative is the third derivative d

^{3}/ dy

^{3}. The order is 3.

## Linearity a Differential Equation

A differential equation is linear if the dependent variable and all its derivative occur linearly in the equation.__Example 2:__Which of these differential equations are linear?

\dfrac{dy}{dx} + x^2 y = x \\\\
\dfrac{1}{x}\dfrac{d^2y}{dx^2} - y^3 = 3x \\\\
\dfrac{dy}{dx} - ln y = 0\\\\
\dfrac{d^3y}{dx^3} - 2 \dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} = 2\sin x

__Solution to Example 2__

1. Both dy/dx and y are linear. The differential equation is linear.

2. The term y

^{3}is not linear. The differential equation is not linear.

3. The term ln y is not linear. This differential equation is not linear.

4. The terms d

^{3}y / dx

^{3}, d

^{2}y / dx

^{2}and dy / dx are all linear. The differential equation is linear.

__Example 3:__

General form of the first order linear differential equation.

\dfrac{dy}{dx}+P(x) y = Q(x)

__Example 4:__

General form of the second order linear differential equation.

\dfrac{d^2y}{dx^2}+P(x)\dfrac{dy}{dx} + Q(x)y = R(x)

__Exercises:__ Determine the order and state the linearity of each differential below.

(\dfrac{d^3y}{dx^3})^4 + 2\dfrac{dy}{dx} = \sin x \\
\dfrac{dy}{dx} - 2x y = x^2- x \\\\
\dfrac{dy}{dx} - \sin y = - x \\\\
\dfrac{d^2y}{dx^2} = 2x y\\\\

__Answers to Above Exercises__

1. order 3 , non linear.

2. order 1 , linear.

3. order 1 , non linear.

4. order 2 , linear.

More references on
Differential Equations

Differential Equations - Runge Kutta Method