Mathematics-IV Unit 2 – Applications of PDE and Fourier Transform: AKTU previous year questions
46 AKTU questions from Unit 2 (Applications of PDE and Fourier Transform) asked in 2021–2026, tagged by marks, year and topic. In short: about 29 marks of every paper come from this unit; the most asked topic is Heat and Wave Equations (37 times); 35 questions came back in a later year. The latest ones are listed below; on the page you can filter them by 2-mark or long questions, topic and repeats.
Unit 2 previous year questions
- Write the one-dimensional heat equation and mention the physical meaning of each term. (2 marks, 2026, Heat and Wave Equations)
- Solve the one-dimensional heat equation ∂u/∂t = c² (∂²u/∂x²), 0 < x < L, t > 0 subject to the boundary conditions u(0,t) = 0, u(L,t) = 0 and the initial condition u(x, 0) = x(L - x) using the method of separation of variables. (7 marks, 2026, Heat and Wave Equations)
- Solve the wave equation ∂²u/∂t² = c² ∂²u/∂x², 0 < x < L subject to u(0,t) = 0, u(L,t) = 0, u(x, 0) = sin(πx/L), ∂u/∂t(x, 0) = 0. (7 marks, 2026, Heat and Wave Equations)
- Using the Fourier transform method, solve the partial differential equation ∂u/∂t = k ∂²u/∂x², -∞ < x < ∞, t > 0 subject to the initial condition u(x, 0) = e^(-a|x|). (7 marks, 2026, Fourier Transform and Inverse)
- How many arbitrary constants in the solution of 1-dimensional wave equation? (2 marks, 2025, Heat and Wave Equations)
- Classify the following operator: t(∂²u/∂t²) + 2(∂²u/∂x∂t) + x(∂²u/∂x²) - (∂u/∂x). (2 marks, 2025, Heat and Wave Equations)
- A tightly stretched flexible string has its ends fixed at x = 0 and x = l. At time t = 0, the string is given a shape defined by F(x) = μx(l − x), μ is a constant and then released. Find the displacement y(x,t) of any point x of the string at any time t > 0. (7 marks, 2025, Heat and Wave Equations)
- Solve by the method of separation of variables: 4(∂u/∂t) + (∂u/∂x) = 3u, u = 3e^(-x) - e^(-5x), when t = 0. (7 marks, 2025, Method of Separation of Variables)
- A bar with insulated sides is initially at a temperature 0°C throughout. The end x = 0 is kept at 0°C, and heat is suddenly applied at the end x = l so that ∂u/∂x = A for x = l, where A is a constant. Find the temperature function u(x,t). (7 marks, 2025, Heat and Wave Equations)
- Solve the Laplace equation ∂²u/∂x² + ∂²u/∂y² = 0, which satisfies the conditions: u(0,y) = u(l,y) = u(x,0) = 0 and u(x,a) = sin(nπx/l). (7 marks, 2025, Heat and Wave Equations)
- Find the Fourier Cosine transform of f(x) = 1/(1 + x²). (7 marks, 2025, Fourier Transform and Inverse)
- A string is stretched and fastened to two points l apart, motion is started by displacing the string into the form y = k(lx – x²) from which it is released at time t=0. Find the displacement of any point on the string at a distance of x from one end at time t. (7 marks, 2025, Heat and Wave Equations)
Most asked Unit 2 topics
- Heat and Wave Equations – asked 37 times
- Method of Separation of Variables – asked 5 times
- Fourier Transform and Inverse – asked 4 times
Unit 2 question pattern
- 2-mark questions: 15 asked in 2021–2026
- 7-mark questions: 16 asked in 2021–2026
- 10-mark questions: 15 asked in 2021–2026
- About 29 marks from Unit 2 in every paper
- 35 questions were asked again in a later year
Unit 2 questions year by year
- 2026: 4 Unit 2 questions asked (23 marks across that year's papers)
- 2025: 8 Unit 2 questions asked (46 marks across that year's papers)
- 2024: 9 Unit 2 questions asked (53 marks across that year's papers)
- 2023: 10 Unit 2 questions asked (68 marks across that year's papers)
- 2022: 10 Unit 2 questions asked (68 marks across that year's papers)
- 2021: 5 Unit 2 questions asked (34 marks across that year's papers)