The GATE Grind

GATE 2022 PH – Question 27

Mathematical Physics · linear differential equations: simple applications of first and second order linear differential equations and solutions · 1 mark · Multiple select

The equation $(1-x^2)y^{\prime\prime}-xy^\prime+9y=0$ has a regular singularity at

  1. -1
  2. 0
  3. +1
  4. no finite x

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) -1; (C) +1

Explanation

Write the equation in standard form $y^{\prime\prime}+P(x)y^\prime+Q(x)y=0$:
$$y^{\prime\prime}-\frac{x}{1-x^2}y^\prime+\frac{9}{1-x^2}y=0 .$$

$P(x)=-\dfrac{x}{1-x^2}$ and $Q(x)=\dfrac{9}{1-x^2}$ are singular where $1-x^2=0$, i.e. at $x=\pm1$. At $x=0$ they are finite, so $x=0$ is an ordinary point.

**Regular singular point test:** $(x-x_0)P$ and $(x-x_0)^2Q$ must stay finite.
- At $x_0=1$: $(x-1)P=\dfrac{x(x-1)}{(x-1)(x+1)}\to\dfrac{1}{2}$ and $(x-1)^2Q=\dfrac{9(x-1)^2}{-(x-1)(x+1)}\to0$. Both finite, so $x=1$ is a regular singular point. ✓
- At $x_0=-1$: the same argument holds. ✓

The regular singularities are at $x=-1$ and $x=+1$. Answer **A and C**.