Paper I — Mathematics

Differential Equations

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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Core5 mapped NDA questions

Build the idea

Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.

Concept

A differential equation relates a function to its derivatives. Its order is the highest derivative present; its degree is the power of that highest derivative. First-order equations are solved by variable separation, homogeneous methods, or the integrating factor.

Growth and decay problems are common applications.

Key Points & Formulas

Order = highest derivative; degree = power of the highest-order derivative.
Variable separable: bring all x with dx and all y with dy, then integrate.
Linear form dy/dx + Py = Q: integrating factor = e^(∫P dx).
Exponential growth/decay: dy/dx = ky ⇒ y = Ce^(kx).

Worked Example

Q. Solve dy/dx = 2xy.

Separate variables: dy/y = 2x dx.

Integrate: ln|y| = x² + c.

y = C e^(x²).

Common traps

Reading the prompt too quickly and missing a qualifier.
Memorising the result without checking when it applies.
Skipping units, signs, chronology or the final reasonableness check.

NDA shortcuts

State the governing rule before substituting values.
Eliminate impossible options before doing full calculation.

One-screen revision sheet

Recall before you practise

Core definition
Most-used rule
One representative example

Past-paper bank

Practise every mapped question

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5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA I 2026

What is the solution of the differential equation $\cos\left(\dfrac{dy}{dx}\right) = p$ when $y(0) = q$ ?

Question 2

NDA II 2025

What is the degree of the differential equation $\footnotesize \left(\dfrac{d^2y}{dx^2}\right)^{3/2} = \left(\dfrac{dy}{dx}\right)^{5/2}$ ?

Question 3

NDA II 2024

What is the solution of the differential equation?

Question 4

NDA I 2024

What are the order and degree respectively of the differential equation {2 − (dy/dx)²}^0.6 = d²y/dx² ?

Question 5

NDA I 2024

If dy/dx = 2e^x y³, y(0)=1/2 then what is 4y²(2−e^x) equal to?

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