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
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?
\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
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.

\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