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