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.

Order and degree

\text{order} = \text{order of the highest derivative present} \qquad \text{degree} = \text{power of the highest-order derivative, after making it polynomial in the derivatives}

Worked example

Find the order and degree of

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

Practice this conceptself-check · 4 quick reps

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.

Degree is undefined when a derivative sits inside a transcendental

For

\dfrac{d^2y}{dx^2} + \sin\!\big(\tfrac{dy}{dx}\big) = 0

the order is 2 but the degree does NOT exist — you can never make it polynomial in

\tfrac{dy}{dx}

. Writing "degree 1" because you see a first power is the trap; a derivative inside

\sin

\cos

\ln

e^{(\cdot)}

kills the degree.

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}

Practice this conceptself-check · 3 quick reps

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.

Formulas (1)

Order and degree of a differential equation
Order and degree
$\text{order} = \text{order of the highest derivative present} \qquad \text{degree} = \text{power of the highest-order derivative, after making it polynomial in the derivatives}$

Watch out for (3)

Clear fractional powers BEFORE reading the degree
→ Order and degree of a differential equation
Degree is undefined when a derivative sits inside a transcendental
→ Order and degree of a differential equation
Count INDEPENDENT constants
→ Solutions and arbitrary constants

Drill every past-year question on this subtopic

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

Drill the 22 questions