Practice tests · GATE

GATE differential equations: 5 practice questions

By the PrepMatic team · Updated 9 Oct 2026 · 4 min read

Differential equations are part of the maths block in EC, EE, ME, CE and RA, but not CS, whose maths is discrete maths, linear algebra, calculus and probability. The questions mix three skills: solve a first or second-order ODE, use Laplace transforms, and recognise a solution of a partial differential equation. The five original questions below practise each of them.

What GATE tests in differential equations

The 2027 syllabi list differential equations for these branches:

• EC: differential equations including partial differential equations.
• EE: differential equations alongside complex variables and probability.
• ME: differential equations including Laplace transforms and the heat, wave and Laplace equations.
• CE: ordinary differential equations, plus partial differential equations (Fourier series, diffusion, wave and Laplace equations).
• RA: differential equations with linear algebra, calculus and numerical methods.

Check your own paper's exact list on the official syllabus page. The standard question types are first-order linear and separable equations, second-order linear equations with constant coefficients and Cauchy-Euler equations, initial value problems, Laplace transforms and their inverses, and quick checks of whether a function satisfies a PDE. All five below are MCQ; the real paper also uses MSQ and NAT.

Practice questions

Question 1

The solution of dy/dx + 2y = e^(−x) with y(0) = 0 is y(x). The value of y(ln 2) is:

• A. 1/4
• B. 1/2
• C. 3/4
• D. 1/8

Question 2

The solution of y″ − 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 0 has the value y(1) equal to:

• A. e²
• B. −e²
• C. −2e²
• D. 0

Question 3

A particular integral of y″ + 4y = cos 2x is:

• A. (x/2) sin 2x
• B. (1/4) cos 2x
• C. −(x/4) sin 2x
• D. (x/4) sin 2x

Question 4

The inverse Laplace transform of F(s) = 1 / [s(s + 3)] is:

• A. 1 − e^(−3t)
• B. (1 + e^(−3t)) / 3
• C. (1 − e^(−3t)) / 3
• D. (e^(−3t) − 1) / 3

Question 5

Which one of the following functions satisfies the one-dimensional wave equation ∂²u/∂t² = 4 ∂²u/∂x²?

• A. u = sin(x + 2t)
• B. u = sin(x + 4t)
• C. u = sin(2x + t)
• D. u = sin(x + t)

Answers and explanations

1. A. The integrating factor is e^(2x), so d/dx (y e^(2x)) = e^(2x) e^(−x) = e^x. Integrating gives y e^(2x) = e^x + C, and y(0) = 0 gives C = −1. So y = e^(−x) − e^(−2x). At x = ln 2, e^(−x) = 1/2 and e^(−2x) = 1/4, so y = 1/2 − 1/4 = 1/4.
2. B. The characteristic equation (m − 2)² = 0 has a repeated root, so y = (A + Bx) e^(2x). The condition y(0) = 1 gives A = 1. Since y′ = (B + 2A + 2Bx) e^(2x), the condition y′(0) = 0 gives B + 2A = 0, so B = −2. Then y(1) = (1 − 2) e² = −e².
3. D. The right-hand side cos 2x is a solution of the homogeneous equation (the roots are ±2i), so resonance occurs and the particular integral has an extra factor of x. Using the operator rule, 1/(D² + 4) applied to cos 2x gives (x/4) sin 2x. Check: for y = (x/4) sin 2x, y″ = cos 2x − x sin 2x, so y″ + 4y = cos 2x.
4. C. Partial fractions: 1/[s(s + 3)] = (1/3)[1/s − 1/(s + 3)]. The inverse transform is (1/3)(1 − e^(−3t)). A quick check is the final value theorem: lim s F(s) as s → 0 equals 1/3, which matches the limit of (1 − e^(−3t))/3 as t → ∞.
5. A. A function f(x + ct) satisfies u_tt = c² u_xx. Here c² = 4, so c = 2 and sin(x + 2t) works: u_tt = −4 sin(x + 2t) and 4 u_xx = −4 sin(x + 2t). For sin(x + 4t) the left side is 16 times the sine but the right side only 4 times; for sin(2x + t) the left side is 1 times and the right is 16; for sin(x + t), 1 against 4.

How to prepare differential equations

• Sort every ODE by type before you touch it: separable, linear first-order (integrating factor), exact, homogeneous, Bernoulli, constant-coefficient linear, or Cauchy-Euler. Naming the type picks the method.
• For constant-coefficient equations, use the characteristic equation for the complementary function and the operator method for the particular integral. Learn the resonance rule from Question 3: if the forcing term already solves the homogeneous equation, multiply by x.
• Memorise a short Laplace table: 1, tⁿ, e^(at), sin(at), cos(at) and the shift, derivative and integral rules, then practise partial fractions for inverses.
• For PDEs, learn the three standard equations (heat, wave, Laplace), the form of their separable solutions and the meaning of the constant c, and practise testing whether a given function satisfies one.
• Always apply initial conditions at the end, not the beginning. Writing the general solution first avoids sign errors in the constants.

Differential equations link directly to signals and systems, control systems and circuits, so work through the GATE engineering mathematics guide for your branch and then try the signals and systems page, where Laplace transforms reappear. Electrical aspirants can see how these topics sit in the GATE electrical preparation guide.

Take the full topic test on PrepMatic

PrepMatic's topic tests run about 15 questions in 30 minutes, which is a good way to check whether you can finish a set of ODE problems on the clock. The free assessment shows your weakest subjects first.

← All practice tests