GATE 2023 MA – Question 62

Ordinary Differential Equations · Series Solutions and Special Functions · 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 $\frac12\int_{-1}^1\sum_{k=1}^n(y_k-y_{k-1})^2dx-\int_{-1}^1\sum_{k=1}^ny_k^2dx=24$, find the positive integer n.

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Correct answer: 12 to 12

Explanation

These are y_k=5P_k, where Legendre polynomials are orthogonal and ‖P_k‖²=2/(2k+1). Cross terms vanish, and the difference of sums telescopes to (1/2)(‖y₀‖²−‖y_n‖²)=25−25/(2n+1). Equate to 24: 2n+1=25, hence **n=12**.

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