GATE Numerical Analysis: Numerical Initial Value Problems – Previous Year Questions

5 GATE previous year questions on Numerical Initial Value Problems (Numerical Analysis, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q41 (2 marks, Multiple choice) – Explicit Euler solves y′=−10y,t>0,y(0)=1 via y (n+1)=y n+h f(t n,y n). Which step range ensures y n→0?
2. GATE 2022 MA Q58 (2 marks, Numerical answer) – For y′=f(x,y), a second-order Runge–Kutta method uses K₁=hf(x i,y i),K₂=hf(x i+αh,y i+βK₁), y (i+1)=y i+(K₁+3K₂)/4. Find αβ to two decimals.
3. GATE 2024 MA Q22 (1 mark, Multiple choice) – For y′=f(x,y), use y n=y (n−1)+ak₁+bk₂, with k₁=hf(x (n−1),y (n−1)), k₂=hf(x (n−1)+αh,y (n−1)+βk₁). Which (a,b,α,β) gives order two?
4. GATE 2025 MA Q35 (1 mark, Numerical answer) – Use Euler with h=0.5 for y′=sin(π(x+y)), y(0)=0. Find approximate y(1.5)+2y(1) (integer).
5. GATE 2026 MA Q30 (1 mark, Numerical answer) – For y′ = x+y, y(0) = 1, use modified Euler with h = 0.05. Find the second approximation to y(h) to two decimals.