Order and Linearity of Differential Equations
A tutorial on how to determine the order and linearity of a differential equations.
Order of a Differential EquationThe 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} +(1x)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 EquationA 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:
\dfrac{dy}{dx}+P(x) y = Q(x)
Example 4:
\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
