Let α,β be Frobenius indicial roots at zero for 2x2y-xy+(1-x2)y=0. Which statements hold?

GATE 2026 MA – Question 51

Ordinary Differential Equations · Series Solutions and Special Functions · 2 marks · Multiple select

Let α,β be Frobenius indicial roots at zero for $2x^2y^{\prime\prime}-xy^\prime+(1-x^2)y=0$. Which statements hold?

α≠β
α−β∈ℤ
α−β∉ℤ
α=β

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Show answer and explanation

Correct answer: (A) α≠β; (C) α−β∉ℤ

Explanation

Insert leading term xʳ and retain the lowest power: $2r(r-1)-r+1=0$, or $(2r-1)(r-1)=0$. Roots are 1/2 and 1, distinct with difference ±1/2, not an integer. **A,C**.

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