GATE Partial Differential Equations: Second Order Equations and Laplace Equation – Previous Year Questions

10 GATE previous year questions on Second Order Equations and Laplace Equation (Partial Differential Equations, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q37 (2 marks, Multiple choice) – For u xx+4u xy+(x²+4y²)u yy=sin(x+y), P:the PDE is parabolic on x²/4+y²=1. Q:it is hyperbolic inside that ellipse. Choose truth values.
2. GATE 2022 MA Q35 (1 mark, Numerical answer) – Transform 7u xx+16u xy+4u yy=0 using ξ=y−2x,η=7y−2x into Au ξξ+Bu ξη+Cu ηη=0. Find (B²−4AC)/12³.
3. GATE 2022 MA Q61 (2 marks, Numerical answer) – A harmonic u on 1<x<2,1<y<2 has u x(1,y)=y,u x(2,y)=5,u y(x,1)=αx²/7,u y(x,2)=x. If a solution exists, find α.
4. GATE 2023 MA Q61 (2 marks, Numerical answer) – On Ω=x²+y²<4, u xx+u yy=0 and u=1+2x² on ∂Ω. Find u(0,1).
5. GATE 2024 MA Q46 (2 marks, Multiple choice) – On x,y>0, transform y²u xx−x²u yy+yu y=0 using η=x²/2, ξ=y²/2. It becomes (u -u )+(u +u )/(2)+(a u +b u )/(2)=0. Find (a,b).
6. GATE 2024 MA Q65 (2 marks, Numerical answer) – In the upper half-disk 0<r<2,0<θ<π, u rr+r⁻¹u r+r⁻²u θθ=0, with u(r,0)=u(r,π)=0 for 0≤r≤2 and u(2,θ)=k sin2θ. Given u(1,π/4)=2, find k to two…
7. GATE 2025 MA Q19 (1 mark, Multiple choice) – Classify (1+x2)u xx+2x(1-y2)u xy+(1-y2)u yy+xu x+(1-y2)u y=0.
8. GATE 2026 MA Q25 (1 mark, Multiple select) – For distinct a,b > 0, consider af xx+bf yy=8f. Which pairs (ξ,η) make f=e2 x+ y a solution?
9. GATE 2026 MA Q44 (2 marks, Multiple choice) – T is harmonic in 0 < x,y < 1, with T(x,0)=x, T(0,y)=y, T(x,1)=1+x, T(1,y)=1+y. Find T(1/2,1/3).
10. GATE 2026 MA Q61 (2 marks, Numerical answer) – u is harmonic in the open unit disk and equals x²−y² on its boundary. Find 4[u(1/2,0)-u(0,1/2)] (integer).