# 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

\(
1) \dfrac{dy}{dx} + y^2 x = 2x \\\\
2) \dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} + y = 0 \\\\
3) 10 y'' - y = e^x \\\\
4) \dfrac{d^3}{dx^3} - x\dfrac{dy}{dx} +(1-x)y = \sin y
\)
__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 \( d^2y / dx^2 \), 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?

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

__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.

\(
1) (\dfrac{d^3y}{dx^3})^4 + 2\dfrac{dy}{dx} = \sin x \\
2) \dfrac{dy}{dx} - 2x y = x^2- x \\\\
3) \dfrac{dy}{dx} - \sin y = - x \\\\
4) \dfrac{d^2y}{dx^2} = 2x y\\\\
\)
__Answers to Above Exercises__

1. order 3 , non linear.

2. order 1 , linear.

3. order 1 , non linear.

4. order 2 , linear.

## References and Links

Differential Equations

Differential Equations - Runge Kutta Method