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 d2y / 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 d3 / 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 y3 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 d3y / 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.
