GATE Ordinary Differential Equations: Series Solutions and Special Functions – Previous Year Questions

9 GATE previous year questions on Series Solutions and Special Functions (Ordinary Differential Equations, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q60 (2 marks, Numerical answer) – A power-series solution y=Σₖ₌₀∞a kxᵏ,a₀≠0, satisfies y″−24x²y=0. Find a₄/a₀.
2. GATE 2022 MA Q33 (1 mark, Numerical answer) – If x²φ″+xφ′+x²φ=0,x>0 has a Frobenius solution =xr n0a nxn with a₀≠0, find r²+1.
3. GATE 2022 MA Q34 (1 mark, Numerical answer) – Bessel functions satisfy J (α−1)(x)+J (α+1)(x)=(2α/x)J α(x). Find (πJ (3/2)(π))².
4. GATE 2022 MA Q52 (2 marks, Multiple select) – An orthonormal set in L²[−1,1] has φ n=C nP n, where P n is the degree-n Legendre polynomial. Which listed statements hold?
5. GATE 2023 MA Q62 (2 marks, Numerical answer) – For k≥0, y k is a degree-k polynomial satisfying y k(1)=5 and (1−x²)y k″−2xy k′+k(k+1)y k=0. If 12 -11 k=1n(y k-y k-1)2dx- -11 k=1ny k2dx=24, find the…
6. GATE 2023 MA Q63 (2 marks, Numerical answer) – For 4(ln x)y″+3y′+y=0,x>1, let r₁,r₂ be the indicial roots at regular singular point x=1. Find r₁−r₂ to two decimals.
7. GATE 2025 MA Q28 (1 mark, Numerical answer) – Legendre P n satisfies (1−x²)y″−2xy′+n(n+1)y=0, with P n(1)=1. Find P₁₁′(1) (integer).
8. GATE 2026 MA Q27 (1 mark, Multiple select) – For 2x(x-2)2y+3xy+(x-2)y=0, which statements about x = 2 are true?
9. GATE 2026 MA Q51 (2 marks, Multiple select) – Let α,β be Frobenius indicial roots at zero for 2x2y-xy+(1-x2)y=0. Which statements hold?