Order and Degree of a Differential Equation

What Is The Order And Degree Of A Differential Equation?

The order and degree of a differential equation are two separate labels that describe its highest derivative. A differential equation relates a function to its derivatives, and these two numbers are the first things you read off it before choosing a solution method.

• Order is the order of the highest derivative present. If the highest derivative is $\frac{d^2y}{dx^2}$, the order is $2$; if it is $\frac{d^3y}{dx^3}$, the order is $3$.

• Degree is the power to which that highest-order derivative is raised, but only after the equation has been written as a polynomial in its derivatives: no radicals over derivatives, and no derivatives in a denominator.

Take the equation

$$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0.$$

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$. That derivative appears to the first power and nothing needs clearing, so the degree is $1$. Every equation on this page sits inside the wider family of ordinary differential equations, and reading its order and degree is the classification step that decides everything after it.

How Do You Find The Order Of A Differential Equation?

To find the order, locate the highest derivative in the equation and report its order. Nothing else on the page matters for this step: not the power on the derivative, not the power on $y$, and not how many times a lower derivative appears.

• In $\frac{dy}{dx} + y = x$, the highest derivative is the first derivative, so the order is $1$.

• In $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^5 + y = 0$, the highest derivative is the second derivative, so the order is $2$. The power $5$ on the first derivative changes nothing here, because $\frac{dy}{dx}$ is not the highest derivative.

• In $\frac{d^3y}{dx^3} + y\frac{dy}{dx} = \sin x$, the highest derivative is the third derivative, so the order is $3$.

The order is always a whole number, and it never depends on any exponent. It counts how deep the derivatives go, which is why it is the easier of the two numbers to read.

How Do You Find The Degree Of A Differential Equation?

To find the degree, first rewrite the equation as a polynomial in its derivatives, then read the power on the highest-order derivative. The rewriting step is where marks are won or lost, so treat it as compulsory.

There are two things to clear before the equation is a polynomial in its derivatives:

1. Radicals over derivatives. A square root or fractional power containing a derivative must be removed by raising both sides to a power.

2. Derivatives in a denominator. A derivative in the bottom of a fraction must be cleared by multiplying through.

Only once both are gone do you read the exponent on the highest-order derivative. That exponent is the degree.

The degree, when it exists, is always a positive whole number. It answers a different question from the order: not how deep the derivatives go, but to what power the deepest one is raised once the equation is honest polynomial in its derivatives.

What Are Some Worked Examples Of Order And Degree?

Each example is read the same way: find the highest derivative for the order, then clear radicals and fractions of derivatives before reading the degree.

Example 1: Find the order and degree of $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$.

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$. It appears to the first power, and there is no radical or fraction to clear, so the degree is $1$.

Final answer: order $2$, degree $1$.

Example 2: Find the order and degree of $\left(\frac{dy}{dx}\right)^3 + y = x$.

The highest derivative is $\frac{dy}{dx}$, so the order is $1$. The equation is already a polynomial in that derivative, which is raised to the power $3$, so the degree is $3$.

Final answer: order $1$, degree $3$.

Example 3: Find the order and degree of $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2}$.

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$. As written, that derivative sits alone on the right and looks like a first power, but the left side still contains a radical over a derivative, so the equation is not yet a polynomial in its derivatives. Square both sides to clear it:

$$1 + \left(\frac{dy}{dx}\right)^2 = \left(\frac{d^2y}{dx^2}\right)^2.$$

Now the equation is a polynomial in its derivatives, and the highest-order derivative $\frac{d^2y}{dx^2}$ appears squared. The degree is $2$, not $1$. Reading it before squaring is the classic error this example is built to expose.

Final answer: order $2$, degree $2$.

Example 4: Find the order and degree of $\frac{d^2y}{dx^2} + \frac{1}{\frac{dy}{dx}} = 0$.

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is $2$. A derivative sits in a denominator, so the equation is not a polynomial in its derivatives yet. Multiply through by $\frac{dy}{dx}$:

$$\frac{dy}{dx}\cdot\frac{d^2y}{dx^2} + 1 = 0.$$

The highest-order derivative $\frac{d^2y}{dx^2}$ now appears to the first power, so the degree is $1$.

Final answer: order $2$, degree $1$.

When Is The Degree Of A Differential Equation Not Defined?

The degree is not defined when the equation can never be made a polynomial in its derivatives. This happens whenever a derivative sits inside a transcendental function, such as a sine, cosine, exponential, or logarithm, because no amount of algebra removes it.

Compare two equations that look alike:

• $\frac{dy}{dx} = \sin(x + y)$ has degree $1$. The sine wraps $x + y$, not a derivative, so the highest derivative $\frac{dy}{dx}$ still appears to the first power.

• $\sin\left(\frac{dy}{dx}\right) = x + y$ has no defined degree. The sine wraps the derivative itself. Expanding it as an infinite series would give $\frac{dy}{dx}$ raised to every odd power at once, so there is no single highest power to name.

The example the contract-scope flags is the same idea one order up:

$$\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0.$$

The highest derivative is $\frac{d^2y}{dx^2}$, so the order is still $2$. But $\frac{dy}{dx}$ is trapped inside a sine, so the equation is not a polynomial in its derivatives and the degree is not defined. The same verdict holds for $\frac{d^2y}{dx^2} + e^{,dy/dx} = 0$: the order is $2$, and the degree is not defined.

Order is always defined for any differential equation. Degree is a bonus label that exists only for the polynomial-in-derivatives case.

Table: Order, degree, and linearity of common differential equations.

What Is The Difference Between Order, Degree, And Linearity?

Order and degree are often confused with a third label, linearity, and the three are genuinely independent. Keeping them apart removes most of the difficulty in this topic.

• Order counts the deepest derivative.

• Degree counts the power on that deepest derivative, after clearing radicals and fractions of derivatives.

• Linearity is stricter than either: an equation is linear when $y$ and all its derivatives appear only to the first power, are never multiplied together, and carry coefficients that depend on $x$ alone.

The trap is assuming degree $1$ means linear, and it does not. The equation $\frac{dy}{dx} + y^2 = 0$ has degree $1$, because its highest derivative $\frac{dy}{dx}$ is to the first power, yet it is nonlinear, because $y$ appears squared. A linear differential equation always has degree $1$, but a degree-$1$ equation need not be linear, since degree looks only at the highest derivative while linearity looks at the whole equation.

Why Do Order And Degree Matter?

These two labels are not bookkeeping. Each one predicts something concrete about the solutions before you solve anything.

• The order tells you how many arbitrary constants the general solution carries. A first-order equation has one constant $C$; a second-order equation has two. Geometrically, the solutions form a family of curves, and the order is the number of independent dials, so it is exactly how many conditions, such as an initial value problem, you need to pin down one curve.

• The degree tells you how many slope branches the equation allows. Once the equation is a polynomial of degree $n$ in the highest derivative, solving for that derivative can give up to $n$ values at a point. A degree-$1$ equation assigns a single direction through each point, which is why its direction field has one clean flow; a higher degree can carry several.

• Together they choose the method. Whether an equation is separable, needs an integrating factor, or requires a numerical scheme depends first on its order and degree. Read them wrong and every step after is aimed at the wrong technique.

That is the payoff of the classification. The algebra you read off the symbols (highest derivative, its power) matches the geometry of the solution family (how many constants, how many directions), so the two numbers are a forecast, not a formality.

Who Discovered The Order And Degree Of A Differential Equation?

The idea of sorting differential equations by their highest derivative grew alongside calculus itself, as soon as there were enough equations to need organising.

Two more figures shaped what these numbers mean:

• Gottfried Wilhelm Leibniz (1646–1716, Germany) introduced the $\frac{dy}{dx}$ notation that lets the order and degree be read directly from an equation.

• Leonhard Euler (1707–1783, Switzerland) developed the general theory of ordinary differential equations and made precise the link between the order of an equation and the number of arbitrary constants its solution carries.

Where Are Order And Degree Used In The Real World?

Because the order fixes how many conditions a model needs, and the degree flags how tangled its rates are, engineers and scientists read both before trusting a model.

• Motion and mechanics: Newton's second law is second-order in position, so a moving object needs two starting facts, its position and its velocity, which is exactly what the order predicts.

• Electrical circuits: a resistor-inductor-capacitor circuit is modelled by a second-order equation, and its order tells the engineer how many independent initial measurements set the current.

• Population and finance: simple growth and decay models are first-order, needing only a single starting amount, while models with acceleration or feedback rise in order.

• Control and robotics: the order of the governing equation decides how many sensors must report state, and a high degree warns that the response can branch, so designers keep systems low-degree where they can.

• Structural engineering: beam-bending is famously fourth-order, and knowing the order up front tells a designer how many boundary conditions the beam's ends must supply.

One classification, read in seconds, tells a modeller how much starting information a real system demands and how complicated its behaviour may get.

What Are The Most Common Mistakes With Order And Degree Of A Differential Equation?

These three errors account for most lost marks, and each matches a question real students ask on Quora, r/learnmath, and NCERT practice forums.

Reading the degree before clearing radicals or fractions of derivatives.

Where it slips in:

A student sees $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2}$, notices $\frac{d^2y}{dx^2}$ to the first power, and writes degree $1$.

Don't do this:

Do not read any power while a derivative is under a root or in a denominator. The equation is not yet a polynomial in its derivatives.

The correct way:

Clear the radical first. Squaring gives $\left(\frac{d^2y}{dx^2}\right)^2 = 1 + \left(\frac{dy}{dx}\right)^2$, so the true degree is $2$.

Taking the power of $y$ instead of the highest derivative.

Where it slips in:

Faced with $\left(\frac{dy}{dx}\right)^2 + y^3 = 0$, a student reports degree $3$ by reading the exponent on $y$.

Don't do this:

Do not let the power on $y$ or on any lower term set the degree. Only the highest-order derivative counts.

The correct way:

Find the highest derivative, here $\frac{dy}{dx}$, and read its power. The order is $1$ and the degree is $2$; the $y^3$ term is irrelevant to both.

Claiming a degree for an equation with a derivative inside a transcendental function.

Where it slips in:

A student writes a degree for $\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$ by guessing a power for the sine term.

Don't do this:

Do not force a degree onto an equation that cannot be made a polynomial in its derivatives. A derivative inside $\sin$, $\cos$, $e^{(\cdot)}$, or $\log$ blocks it.

The correct way:

Report the order as usual, here $2$, and state that the degree is not defined. That is the complete, correct answer.

Practice Problems On Order And Degree Of A Differential Equation

Read the order first, then clear radicals and fractions of derivatives before reading the degree. Verified answers follow each line.

1. Find the order and degree of $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^4 + y = 0$.
   (Answer: order $3$, degree $1$. The highest derivative $\frac{d^3y}{dx^3}$ is to the first power; the power $4$ sits on a lower derivative.)

2. Find the order and degree of $\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} + y = 0$.
   (Answer: order $2$, degree $3$.)

3. Find the order and degree of $\frac{d^2y}{dx^2} = \sqrt[3]{,\frac{dy}{dx} + 1,}$.
   (Answer: cube both sides to get $\left(\frac{d^2y}{dx^2}\right)^3 = \frac{dy}{dx} + 1$; order $2$, degree $3$.)

4. Find the order and degree of $\frac{dy}{dx} + \cos\left(\frac{dy}{dx}\right) = 0$.
   (Answer: order $1$, degree not defined; the derivative sits inside a cosine.)

5. Is $\frac{dy}{dx} + xy = x^2$ linear, and what is its degree?
   (Answer: linear, degree $1$; $y$ and $\frac{dy}{dx}$ each appear to the first power with $x$-only coefficients.)

6. Find the order and degree of $y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0$.
   (Answer: order $2$, degree $1$; the highest derivative is to the first power, though the product $y\frac{d^2y}{dx^2}$ makes it nonlinear.)

Where Should You Go Next After Order And Degree Of A Differential Equation?

Order and degree are the classification step, and several natural doors open once you can read them.

1. Ordinary differential equations. The full family these labels classify, and how the order shapes each equation's general solution.

2. Separable differential equations. The first solving method most first-order, degree-one equations meet.

3. Exact differential equations. Another first-order technique, plus the test that tells you when it applies.

4. Initial value problems. Use the order to count how many conditions turn a general solution into a specific one.

If your child is meeting differential equations for the first time, a live Bhanzu trainer teaches order and degree from the picture up, so the classification feels like a forecast rather than a rule to memorise, in the Bhanzu math program.