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
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 d2y / 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 d3 / 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?
Solution to Example 2
1. Both dy/dx and y are linear. The differential equation is linear.
2. The term y3 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 d3y / dx 3, d2y / dx 2 and dy / dx are all linear. The differential equation is linear.
General form of the first order linear differential equation.
General form of the second order linear differential equation.
Exercises: Determine the order and state the linearity of each differential below.
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 - Runge Kutta Method