Order and Degree

A differential equation is an equation that relates a function to one or more of its derivatives. Before solving any differential equation, mathematicians classify it using two key properties: its order and its degree. These classifications immediately reveal how complex the equation is and which solution techniques are likely to work. The order of a differential equation is determined by the highest-order derivative that appears in the equation. For example, if the highest derivative present is a second derivative, the equation is called a second-order differential equation. This is the most fundamental classification because it tells you the minimum number of initial or boundary conditions needed to pin down a unique solution. The degree of a differential equation is the power (exponent) to which the highest-order derivative is raised, provided the equation can be written as a polynomial in its derivatives. Together, order and degree form the basic identity card of any differential equation and are the first things a mathematician checks before attempting a solution.

Key takeaways
  • Identify the Derivatives Present
  • Find the Order
  • Prepare to Find the Degree
  • Find the Degree
  • Special Cases and Common Pitfalls

Identify the Derivatives Present

List every derivative in the equation. Use standard notation: y' or dy/dx for the first derivative, y'' or d²y/dx² for the second derivative, y''' for the third, and so on. For example, in the equation y'' + 3y' − 2y = sin(x), the derivatives present are y' (first) and y'' (second).

Find the Order

Look at the highest-order derivative you listed. Its order number is the order of the entire differential equation. In the example above, y'' is a second derivative, so the equation is second-order. A general rule: an nth-order ordinary differential equation requires n arbitrary constants in its general solution and therefore n conditions to find a particular solution.

Prepare to Find the Degree

To find the degree, the equation must first be expressed as a polynomial in its derivatives — that means no derivatives appear inside radicals, logarithms, trigonometric functions, or exponentials. If you have an equation like \sqrt{y''} + y = 0, rewrite it by squaring both sides: (y'')^1 after squaring gives degree 2. If the equation cannot be made polynomial in its derivatives (e.g., e^{y'} = x), the degree is said to be undefined or not defined.

Find the Degree

Once the equation is in polynomial form in its derivatives, identify the power to which the highest-order derivative is raised. This power is the degree. For example: (y'')^3 + 2(y')^5 − y = 0 has order 2 (highest derivative is y'') and degree 3 (y'' is raised to the power 3). Note that the powers of lower-order derivatives do not affect the degree — only the power of the highest-order derivative matters.

Special Cases and Common Pitfalls

Students often confuse the order and degree, or accidentally assign the degree based on a lower-order derivative's power. Remember: Order = which derivative is highest; Degree = what power that highest derivative is raised to. Also, degree is only defined for polynomial differential equations. Non-linear DEs where the highest derivative appears in a non-polynomial way (inside sin, cos, e^x, etc.) have an undefined degree. Additionally, a first-order, first-degree equation (e.g., dy/dx = f(x,y)) is the simplest category and is usually the starting point in any differential equations course.

The Key Equation

an(x)dnydxn+an−1(x)dn−1ydxn−1+⋯+a1(x)dydx+a0(x)y=g(x)a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)
  • n — Order (dimensionless (positive integer)): The order of the differential equation; equals the highest derivative present.
  • \frac{d^n y}{dx^n} — nth derivative of y (depends on context): The highest-order derivative in the equation, which determines the order.
  • a_n(x) — Leading coefficient function (depends on context): The coefficient multiplying the highest-order derivative; must be non-zero for the equation to truly be order n.
  • g(x) — Non-homogeneous term (depends on context): The right-hand side function. If g(x) = 0, the equation is called homogeneous.

Worked Example

Problem. Determine the order and degree of each differential equation: (a) \left(\frac{d^2 y}{dx^2}\right)^3 + x\frac{dy}{dx} - y = 0, (b) \frac{d^3 y}{dx^3} + \sin\!\left(\frac{dy}{dx}\right) = x, (c) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2 y}{dx^2}

(a) Order 2, Degree 3;(b) Order 3, Degree undefined;(c) Order 2, Degree 2\text{(a) Order }2,\text{ Degree }3;\quad \text{(b) Order }3,\text{ Degree undefined};\quad \text{(c) Order }2,\text{ Degree }2

Answer. Part (a): order 2, degree 3. Part (b): order 3, degree undefined. Part (c): order 2, degree 2.

Figure
Diagram
Classification of Differential Equations by Order and DegreeOrder = power of the HIGHEST derivative presentDegree = power to which that highest derivative is RAISED (if polynomial)1st Order, 1st Degreedy/dx + 2y = xHighest deriv: dy/dxPower of dy/dx = 12nd Order, 3rd Degree(y′′)³ + 5y′ = 0Highest deriv: y′′Power of y′′ = 33rd Order, 1st Degreey″ − 3y′ + y = 0Power of y″ = 1Degree Undefinedy′′ + sin(y′) = 0Not polynomial in y′
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently co-invented calculus (alongside Isaac Newton) and introduced the notation dy/dx and d²y/dx² that we still use today to denote derivatives of different orders. His systematic notation made it natural to classify differential equations by the order of their highest derivative. Leibniz was the first to use the term 'differential equation' and worked extensively on first-order equations in the 1690s, laying the groundwork for the formal classification systems students learn today. Leibniz was also a philosopher, diplomat, and librarian. He designed a mechanical calculating machine called the 'Stepped Reckoner' that could multiply and divide — one of the most advanced calculators of the 17th century.

  • Separable Differential Equations — Separable equations are first-order, first-degree DEs — the simplest category in the order-degree classification. Understanding order and degree is essential before tackling solution methods like separation of variables.
  • Linear First-Order Differential Equations — These are order-1, degree-1 equations with a specific linear structure. Identifying the order and degree correctly is the first step in recognising that the integrating factor method applies.
  • Higher-Order Derivatives — The concept of order in differential equations is built directly on knowledge of second, third, and nth derivatives. Without understanding higher-order derivatives, you cannot correctly determine the order of a DE.
  • Homogeneous Differential Equations — The homogeneous/non-homogeneous distinction is the next classification step after order and degree. Together, all these classifications guide which analytical solution method is appropriate.