IIT JAM Differential Equations, Finite Differences & Numerical Analysis: practice questions

IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)

Original practice questions written to the published syllabus · not past papers

10worked here, free

22written to this exam in this subject

11syllabus units

What this subject covers

Ordinary differential equations, interpolation, numerical differentiation and integration and root-finding: what AIEEA-PG Group 07 names explicitly, and what growth models, life tables and iterative estimation (Newton-Raphson for maximum likelihood) rely on.

In the exams: AIEEA-PG Group 07 Units II-III (heavy: differential equations, finite differences, interpolation, numerical integration, Newton-Raphson named); AICE-JRF Computer Application Unit 10 numerical analysis (heavy)

Ordinary differential equations of first order: exact, linear and Bernoulli equations
Higher-order linear equations: complementary function and particular integral
Calculus of finite differences and difference operators
Interpolation: Newton forward and backward, Lagrange and divided differences
Numerical differentiation
Numerical integration: trapezoidal and Simpson's rules
Difference equations
Solution of non-linear equations: bisection and Newton-Raphson
Solution of linear systems: Gaussian elimination and iterative methods
Numerical solution of ODEs: Euler and Runge-Kutta methods
Coordinate geometry of conics

Where the 22 questions fall

How our questions for IIT JAM spread across the units of Differential Equations, Finite Differences & Numerical Analysis. Difficulty: 3 easy, 14 medium, 5 hard.

Ordinary differential equations of first order: exact, linear and Bernoulli equations	3
Higher-order linear equations: complementary function and particular integral	3
Numerical differentiation	2
Numerical integration: trapezoidal and Simpson's rules	2
Difference equations	2
Solution of non-linear equations: bisection and Newton-Raphson	2
Solution of linear systems: Gaussian elimination and iterative methods	2
Numerical solution of ODEs: Euler and Runge-Kutta methods	2
Coordinate geometry of conics	2
Calculus of finite differences and difference operators	1
Interpolation: Newton forward and backward, Lagrange and divided differences	1

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Ordinary differential equations of first order: exact, linear and Bernoulli equations · hard

With K = 1,000, N₀ = 100 and r = 0.5, the logistic solution N(t) = K/(1 + Ae^(−rt)) gives N(2) of about:

Answer and working

C. 232

A = (1,000 − 100)/100 = 9; N(2) = 1,000/(1 + 9e^(−1)) = 1,000/4.311 ≈ 232. 272 uses unchecked exponential growth 100e¹.

Further reading: Boyce, DiPrima and Meade, Elementary Differential Equations (2017); Burden, Faires and Burden, Numerical Analysis (2016); Hildebrand, Introduction to Numerical Analysis (1987); Thornley and France, Mathematical Models in Agriculture (2007)

2 · Higher-order linear equations: complementary function and particular integral · easy

A particular integral of y″ − 3y′ + 2y = e^(3x) is:

e^(3x)/2
e^(3x)/20
xe^(3x)
e^(3x)/6

Answer and working

A. e^(3x)/2

1/f(D) e^(3x) = e^(3x)/f(3) with f(3) = 9 − 9 + 2 = 2.

3 · Higher-order linear equations: complementary function and particular integral · hard

A particular integral of y″ − 4y = e^(2x) is:

e^(2x)/4
e^(2x)
xe^(2x)/4
xe^(2x)/2

Answer and working

C. xe^(2x)/4

f(2) = 0 (resonance); y_p = x e^(2x)/f′(2), and f′(m) = 2m gives 4.

4 · Calculus of finite differences and difference operators · medium

The relation between the shift operator E and the derivative D is:

E = 1 + hD exactly
E = hD
E = log(hD)
E = e^(hD)

Answer and working

D. E = e^(hD)

Taylor's theorem: f(x + h) = Σ hⁿDⁿf/n! = e^(hD)f; 1 + hD is only the first-order part.

5 · Interpolation: Newton forward and backward, Lagrange and divided differences · medium

The Lagrange polynomial through (1, 2), (2, 3) and (4, 11) takes the value at x = 3:

Answer and working

D. 6

The quadratic is x² − 2x + 3; at 3: 9 − 6 + 3 = 6.

6 · Numerical differentiation · medium

The standard approximation of f″(x) from three equally spaced values is:

[f(x + h) − f(x − h)]/(2h)
[f(x + h) − f(x)]/h²
[f(x + h) + 2f(x) + f(x − h)]/h²
[f(x + h) − 2f(x) + f(x − h)]/h²

Answer and working

D. [f(x + h) − 2f(x) + f(x − h)]/h²

It is the second central difference divided by h².

7 · Solution of non-linear equations: bisection and Newton-Raphson · medium

Newton–Raphson for x³ − x − 1 = 0 from x₀ = 1.5 gives x₁ of about:

1.3478
1.3247
1.4000
1.6250

Answer and working

A. 1.3478

f(1.5) = 0.875, f′(1.5) = 5.75; 1.5 − 0.875/5.75 ≈ 1.3478. 1.3247 is the root itself.

8 · Numerical solution of ODEs: Euler and Runge-Kutta methods · medium

For y′ = y, y(0) = 1, h = 0.1, the classical RK4 method gives y₁ = 1.1051708. The error compared with e^0.1 is about:

10⁻³
10⁻²
10⁻⁵
10⁻⁷

Answer and working

D. 10⁻⁷

e^0.1 = 1.1051709; the difference is about 8 × 10⁻⁸.

9 · Numerical solution of ODEs: Euler and Runge-Kutta methods · hard

For y′ = −50y, Euler's method is stable only if the step h is less than:

0.02
0.5
0.04
2

Answer and working

C. 0.04

Stability needs |1 − 50h| < 1, i.e. h < 2/50 = 0.04.

10 · Coordinate geometry of conics · medium

The eccentricity of the hyperbola x²/16 − y²/9 = 1 is:

0.8
1.25
1.33
0.75

Answer and working

B. 1.25

e = √(1 + 9/16) = 5/4.

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