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

differential equation example 1


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 / dx2, 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 / dy3. 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?

differential equation example 2


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
3y / dx 3, d2y / dx 2 and dy / dx are all linear. The differential equation is linear.

Example 3:

General form of the first order linear differential equation.

differential equation example 3

Example 4:

General form of the second order linear differential equation.

differential equation example 4

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

differential equation exercises


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






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