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
\(
1) \dfrac{dy}{dx} + y^2 x = 2x \\\\
2) \dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} + y = 0 \\\\
3) 10 y'' - y = e^x \\\\
4) \dfrac{d^3}{dx^3} - x\dfrac{dy}{dx} +(1-x)y = \sin y
\)
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 \( d^2y / dx^2 \), 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?
\(
1) \dfrac{dy}{dx} + x^2 y = x \\\\
2) \dfrac{1}{x}\dfrac{d^2y}{dx^2} - y^3 = 3x \\\\
3) \dfrac{dy}{dx} - \ln y = 0\\\\
4) \dfrac{d^3y}{dx^3} - 2 \dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} = 2\sin x
\)
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.
\(
1) (\dfrac{d^3y}{dx^3})^4 + 2\dfrac{dy}{dx} = \sin x \\
2) \dfrac{dy}{dx} - 2x y = x^2- x \\\\
3) \dfrac{dy}{dx} - \sin y = - x \\\\
4) \dfrac{d^2y}{dx^2} = 2x y\\\\
\)
Answers to Above Exercises
1. order 3 , non linear.
2. order 1 , linear.
3. order 1 , non linear.
4. order 2 , linear.