NDA Maths · Differential Equations

Order, Degree and Solutions

The order of a differential equation is the highest derivative present; the degree is the power of that highest derivative once the equation is made polynomial in its derivatives; the number of arbitrary constants in a solution equals the order.

All Differential Equations notes NDA Maths strategy

Why this matters

Start here — and bank the easy marks. 22 PYQs, and many simply ask for order and/or degree, which is pure definition once you handle one trap: clear radicals and fractional powers first. The rest connect a solution's arbitrary constants to the order.

Concept 1 of 2

Order and degree of a differential equation

Intuition

Order counts how many times you have differentiated — it is the highest derivative that appears. Degree is the exponent on that highest derivative, but ONLY after you have cleared away any radicals or fractional powers so the equation is polynomial in its derivatives. If a derivative is trapped inside a trig or log, the degree simply does not exist.

Definition

The two classifiers:

Order = the order of the highest derivative present (e.g. $d^2y/dx^2$ gives order 2).
Degree = the power of the highest-order derivative AFTER the equation is made free of radicals and fractional powers (made polynomial in the derivatives).
Degree is undefined when a derivative appears inside a transcendental function, e.g. $\cos\!\big(\tfrac{dy}{dx}\big)$ or $\ln\!\big(\tfrac{dy}{dx}\big)$ .
Tip: $\dfrac{dx}{dy} = \Big(\dfrac{dy}{dx}\Big)^{-1}$ — rewrite mixed derivatives in one form before reading the degree.

Worked example

Find the order and degree of

\Big(\dfrac{d^2y}{dx^2}\Big)^{2} = 1 + \Big(\dfrac{dy}{dx}\Big)^{3}

Highest derivative present is $\dfrac{d^2y}{dx^2}$ → order 2.
The equation is already polynomial in the derivatives (no radicals).
The power of $\dfrac{d^2y}{dx^2}$ is 2 → degree 2.

Answer:Order 2, degree 2.

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the order and degree of

k\dfrac{d^2y}{dx^2} = \Big[1 + \Big(\dfrac{dy}{dx}\Big)^{2}\Big]^{2/3}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Order and degree of $\big(\frac{d^3y}{dx^3}\big)^2 = y^4 + \big(\frac{dy}{dx}\big)^5$ ?
2.
Degree of $\frac{dy}{dx} + \cos\!\big(\frac{dy}{dx}\big) = 0$ ?
3.
After squaring $\big(\frac{d^2y}{dx^2}\big)^{3/2} = \big(\frac{dy}{dx}\big)^{5/2}$ , the degree is?
4.
Order of $x^2\frac{d^3y}{dx^3} - \frac{dy}{dx} = 0$ ?

From the bank · past-year question

Example 1Differential EquationsHARD

The order and degree of the differential equation

k\frac{dy}{dx}=\int\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx

are respectively

[Q74 · Apr · 2020]

Clear fractional powers BEFORE reading the degree

The degree is NOT the fractional exponent you see. For

\big(2-(y')^2\big)^{0.6} = y''

, raise to the 5th power to get

\big(2-(y')^2\big)^3 = (y'')^5

: the degree is 5, not 0.6. Make it polynomial first.

Drill 17 more on order and degree of a differential equation

Concept 2 of 2

Solutions and arbitrary constants

Intuition

A general solution carries one arbitrary constant for each integration — so the number of arbitrary constants equals the order of the equation. Turn that around: to find the order of the ODE behind a given family, just count its independent arbitrary constants.

Definition

Solutions and what they tell you:

A general solution of an order- $n$ ODE contains exactly $n$ arbitrary constants; a particular solution fixes them via conditions.
So the order = number of independent arbitrary constants in the family. $y=a\cos x+b\sin x$ (two constants) → order 2.
An ODE like $\dfrac{d^2y}{dx^2}+k^2y=0$ has periodic (SHM) solutions; $\dfrac{d^2y}{dx^2}-k^2y=0$ gives exponential growth.

Worked example

What is the order of the differential equation whose general solution is

y = c_1 e^{2x} + c_2 e^{-3x}

Count the independent arbitrary constants: $c_1$ and $c_2$ — two of them.
Order = number of arbitrary constants.

Answer:Order 2.

Practice this conceptself-check · 3 quick reps

Try it yourself

What is the order of the differential equation whose solution is

y = a\cos x + b\sin x

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
A general solution has 3 arbitrary constants. The ODE's order is?
2.
Which has periodic solutions: $y''+9y=0$ or $y''-9y=0$ ?
3.
Order of the ODE of all circles with centre on the x-axis (one free constant)?

From the bank · past-year question

Example 2Differential EquationsEASY

What is the order of the differential equation whose solution is

y=a\cos x+b\sin x+ce^{-x}+d

, where a, b, c and d are arbitrary constants?

[Q89 · Sep · 2018]

Count INDEPENDENT constants

y=A[\sin(x+C)+\cos(x+C)]

looks like two constants, but it collapses to

B\sin(x+D)

— still two independent constants, so order 2 (giving

y''+y=0

). Combine first; constants that merge don't each count.

Drill 3 more on solutions and arbitrary constants

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Watch out for (2)

Clear fractional powers BEFORE reading the degree
→ Order and degree of a differential equation
Count INDEPENDENT constants
→ Solutions and arbitrary constants

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Differential EquationsHARD

The order and degree of the DE

y^2=4a(x-a)

, where a is arbitrary, are respectively

[Q91 · Apr · 2018]

Example 2Differential EquationsEASY

What is the order of the differential equation of all ellipses whose axes are along the coordinate axes?

[Q86 · Sep · 2021]

Example 3Differential EquationsEASY

Consider the following for the items that follow: Suppose

E

is the differential equation representing family of curves

y^{2}=2cx+2c\sqrt{c}

where

c

is a positive parameter.

What is the order of the differential equation?

[Q74 · Apr · 2023]

Example 4Differential EquationsMODERATE

Which differential equation has a periodic solution?

[Q88 · Apr · 2018]

Example 5Differential EquationsMODERATE

What are order and degree of

\left(\frac{d^3y}{dx^3}\right)^2=y^4+\left(\frac{dy}{dx}\right)^5

[Q100 · Apr · 2018]

Drill every past-year question on this subtopic

22 questions from the bank — paginated, with cart and Word-export support.

Drill the 22 questions