Order of a Differential Equation

Order of a Differential Equation

The number of the highest derivative in a differential equation. A differential equation of order 1 is called first order, order 2 second order, etc.

Example: The differential equation y" + xy' – x³y = sin x is second order since the highest derivative is y" or the second derivative.

Key Formula

F\!\left(x,\, y,\, y',\, y'',\, \dots,\, y^{(n)}\right) = 0

Where:

$x$ = The independent variable
$y$ = The unknown function of x
$y', y'', \dots$ = The first, second, … derivatives of y with respect to x
$y^{(n)}$ = The nth (highest-order) derivative of y with respect to x
$n$ = The order of the differential equation — the number of the highest derivative present

Worked Example

Problem: Determine the order of the differential equation: y''' + 3y' − 2y = 7x.

Step 1: Identify every derivative of y that appears in the equation.

y''' \quad \text{(third derivative)}, \quad y' \quad \text{(first derivative)}, \quad y \quad \text{(the function itself)}

Step 2: Determine which derivative has the highest number.

y''' = \frac{d^{3}y}{dx^{3}} \quad \Rightarrow \quad \text{third derivative}

Step 3: The order equals the number of that highest derivative.

\text{Order} = 3

Answer: The differential equation is of order 3 (a third-order differential equation).

Another Example

This example highlights the common confusion between the power (exponent) on a derivative and the order of a derivative. The exponent does not affect the order — only the derivative number matters.

Problem: Determine the order of the differential equation: (y'')^5 + 4y' = e^x.

Step 1: List every derivative of y in the equation. Here we see y'' (raised to the 5th power) and y'.

y'' \quad \text{(second derivative)}, \quad y' \quad \text{(first derivative)}

Step 2: Note that the exponent 5 on y'' is a power, not a derivative order. It means (y'')^5, not the fifth derivative.

(y'')^5 \neq y^{(5)}

Step 3: The highest derivative present is y'', the second derivative.

\text{Order} = 2

Answer: The differential equation is of order 2 (second order), despite the exponent of 5.

Frequently Asked Questions

What is the difference between the order and degree of a differential equation?

The order is the number of the highest derivative in the equation (e.g., y''' gives order 3). The degree is the exponent (power) to which that highest-order derivative is raised, after the equation has been cleared of fractions and radicals involving derivatives. For example, in (y'')^5 + y = 0, the order is 2 and the degree is 5.

How do you find the order of a partial differential equation?

The same rule applies: find the derivative with the highest total order of differentiation. For example, in the equation ∂²u/∂x² + ∂²u/∂y² = 0 (Laplace's equation), each term involves a second-order partial derivative, so the order is 2.

Why does the order of a differential equation matter?

The order tells you how many initial or boundary conditions you typically need for a unique solution. A first-order equation requires one condition, a second-order equation requires two, and so on. It also determines which solution methods apply.

Order of a Differential Equation vs. Degree of a Differential Equation

	Order of a Differential Equation	Degree of a Differential Equation
Definition	The number of the highest derivative present	The exponent on the highest-order derivative (after clearing radicals/fractions of derivatives)
Example: (y'')^5 + y' = 0	Order = 2 (highest derivative is y'')	Degree = 5 (y'' is raised to the 5th power)
Always defined?	Yes, for every differential equation	Not always — degree is undefined if the equation cannot be written as a polynomial in derivatives
Effect on solutions	Determines the number of initial/boundary conditions needed	Helps classify linearity and solution complexity

Why It Matters

You encounter the order of a differential equation in every calculus and differential equations course because it is the first thing you identify before choosing a solution method. First-order equations use techniques like separation of variables or integrating factors, while second-order equations often require characteristic equations or variation of parameters. The order also tells you how many initial conditions are needed — for example, a second-order equation modeling a spring-mass system requires both an initial position and an initial velocity.

Common Mistakes

Mistake: Confusing the power (exponent) on a derivative with the order of the derivative.

Correction: The exponent on a derivative term like (y'')^5 does not change the order. The order is still 2 because y'' is the second derivative. The exponent 5 affects the degree, not the order.

Mistake: Counting the number of derivative terms instead of finding the single highest derivative.

Correction: An equation like y''' + y'' + y' = 0 has three derivative terms, but its order is 3 because y''' (the third derivative) is the highest derivative present.

Related Terms

Differential Equation — The broader class of equations being classified
First Order Differential Equation — A differential equation of order 1
Second Order Differential Equation — A differential equation of order 2
nth Derivative — The general derivative whose number n determines order
Second Derivative — The derivative that defines a second-order equation
Degree of a Differential Equation — Often confused with order; measures the exponent instead
Initial Value Problem — Number of initial conditions depends on the order