Order and Linearity of Differential Equations
This tutorial explains how to determine the order and linearity of a differential equation, with worked examples and exercises. See also the introduction to differential equations.
Order of a Differential Equation
The order of a differential equation is defined as the highest derivative appearing in the equation.
Example 1
State the order of each differential equation:
\[ \begin{aligned} 1)&\ \frac{dy}{dx} + x y^2 = 2x \\ 2)&\ \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0 \\ 3)&\ 10y'' - y = e^x \\ 4)&\ \frac{d^3y}{dx^3} - x\frac{dy}{dx} + (1-x)y = \sin x \end{aligned} \]Solution
- 1. Highest derivative is \(dy/dx\) → order 1.
- 2. Highest derivative is \(d^2y/dx^2\) → order 2.
- 3. Highest derivative is \(y''\) → order 2.
- 4. Highest derivative is \(d^3y/dx^3\) → order 3.
Linearity of a Differential Equation
A differential equation is called linear if:
- \(y\) and its derivatives appear only to the first power.
- No products of \(y\) with itself or its derivatives.
- No functions of \(y\) such as \(\sin y\), \(\ln y\), or \(y^2\).
- Coefficients depend only on \(x\).
Example 2
Determine which equations are linear:
\[ \begin{aligned} 1)&\ \frac{dy}{dx} + x^2y = x \\ 2)&\ \frac{1}{x}\frac{d^2y}{dx^2} - y^3 = 3x \\ 3)&\ \frac{dy}{dx} - \ln y = 0 \\ 4)&\ \frac{d^3y}{dx^3} - 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = 2\sin x \end{aligned} \]Solution
- 1. Linear.
- 2. Not linear (contains \(y^3\)).
- 3. Not linear (contains \(\ln y\)).
- 4. Linear.
Standard Forms
First-Order Linear Equation
\[ \frac{dy}{dx} + P(x)y = Q(x) \]Second-Order Linear Equation
\[ \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = R(x) \]Exercises
Determine the order and linearity of each equation:
\[ \begin{aligned} 1)&\ \left(\frac{d^3y}{dx^3}\right)^4 + 2\frac{dy}{dx} = \sin x \\ 2)&\ \frac{dy}{dx} - 2xy = x^2 - x \\ 3)&\ \frac{dy}{dx} - \sin y = -x \\ 4)&\ \frac{d^2y}{dx^2} = 2xy \end{aligned} \]Answers
- 1. Order 3, non-linear.
- 2. Order 1, linear.
- 3. Order 1, non-linear.
- 4. Order 2, linear.