Sturm-Liouville Problems and Special Functions - Ordinary Differential Equations - Mathematics Previous Year Questions

1

2008 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2008

Let \(n \geq 3\) be an integer. Let \(y\) be the polynomial solution of \((1 - x^2) y'' - 2xy' + n(n - 1)y = 0\) satisfying \(y(1) = 1\). Then the degree of \(y\) is

A

\(n\)

B

\(n - 1\)

C

less than \(n - 1\)

D

greater than \(n + 1\)

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2

2009 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2009

Let \( P_n(x) \) be the Legendre polynomial of degree \( n \) such that \( P_n(1) = 1, \ n = 1, 2, \ldots \). If \( \int_{-1}^1 \left( \sum_{j=1}^n \sqrt{j(2j+1)} P_j(x) \right)^2 dx = 20 \), then \( n = \)

A

2

B

3

C

4

D

5

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3

2011 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2011

Let \( y \) be a polynomial solution of the differential equation \( (1 - x^2)y'' - 2xy' + 6y = 0. \) If \( y(1) = 2, \) then the value of the integral \( \int_{-1}^1 y^2 dx \) is

A

1/5

B

2/5

C

4/5

D

8/5

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4

2012 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2012

Let the Legendre equation \( (1-x^{2})y'' - 2xy' + n(n+1)y = 0 \) have \( n^{th} \) degree polynomial solution \( y_{n}(x) \) such that \( y_{n}(1) = 3 \). If \( \int_{-1}^{1} (y_{n}^{2}(x) + y_{n}^{'2}(x)) dx = \frac{144}{15} \), then \( n \) is

A

1

B

2

C

3

D

4

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5

2014 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2014

The boundary value problem, \(\frac{d^2\varphi}{dx^2} + 2\varphi = x; \ \varphi(0) = 0\) and \(\frac{d\varphi}{dx}(1) = 0\), is converted into the integral equation \(\varphi(x) = g(x) + \lambda \int_0^1 k(x,\xi)\varphi(\xi)d\xi\), where the kernel \(k(x,\xi) = \begin{cases} \xi, & 0 < \xi < x \\ x, & x < \xi < 1 \end{cases}\). Then \(g\left(\frac{2}{3}\right)\) is ______________

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6

2016 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2016

Let P_n(x) be the Legendre polynomial of degree n and I = ∫₋₁¹ x^k P_n(x)dx, where k is a non-negative integer. Consider the following statements P and Q:
(P) : I = 0 if k < n.
(Q) : I = 0 if n − k is an odd integer.
Which of the above statements hold TRUE?

A

both P and Q

B

only P

C

only Q

D

Neither P nor Q

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7

2016 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2016

The difference between the least two eigenvalues of the boundary value problem \( y'' + \lambda y = 0, \quad 0 < x < \pi \quad y(0)=0, \quad y'(\pi)=0 \) is equal to ______________

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8

2018 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2018

Let \(p_n(x)\) be the polynomial solution of the differential equation \[ \frac{d}{dx}[(1-x^2)y'] + n(n+1)y = 0 \] with \(p_n(1) = 1\) for \(n = 1, 2, 3, \ldots\). If \[ \frac{d}{dx}[p_{n+2}(x) - p_n(x)] = \alpha_n p_{n+1}(x), \] then \(\alpha_n\) is

A

2n

B

2n + 1

C

2n + 2

D

2n + 3

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9

2019 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2019

Consider the boundary value problem (BVP) \( \frac{d^{2} y}{d x^{2}}+\alpha y(x)=0 \), \( \alpha \in \mathbb{R} \) (the set of all real numbers), with the boundary conditions \( y(0)=0, y(\pi)=k \) (\( k \) is a non-zero real number). Then which one of the following statements is TRUE?

A

For \(\alpha=1 \), the BVP has infinitely many solutions

B

For \(\alpha=1 \), the BVP has a unique solution

C

For \(\alpha=-1, k<0 \), the BVP has a solution \(y(x) \) such that \(y(x)>0 \) for all \(x \in(0, \pi) \)

D

For \(\alpha=-1, k>0 \), the BVP has a solution \(y(x) \) such that \(y(x)>0 \) for all \(x \in(0, \pi) \)

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10

2019 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2019

Let \(y : [-1, 1] \to \mathbb{R}\) with \(y(1) = 1\) satisfy the Legendre differential equation \((1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 6y = 0\) for \(|x| < 1\).
Then the value of \(\int_{-1}^1 y(x) (x + x^2) dx\) is equal to ______ (round off to 2 places of decimal).

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11

2020 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2020

For \( n \in \mathbb{N} \cup \{0\} \), let \( y_n \) be a solution of the differential equation
\( xy'' + (1 - x)y' + ny = 0 \)
satisfying \( y_n(0) = 1 \). For which of the following functions \( w(x) \), the integral
\( \int_0^\infty y_p(x) y_q(x) w(x) dx, \quad (p \neq q) \)
is equal to zero?

A

\(e^{-x^2} \)

B

\(e^{-x} \)

C

\(x e^{-x^2} \)

D

\(x e^{-x} \)

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12

2021 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2021

The eigenvalues of the boundary value problem
\[\frac{d^2y}{dx^2} + \lambda y = 0, \quad x \in (0, \pi), \lambda > 0,\]
\[y(0) = 0, \quad y(\pi) - \frac{dy}{dx}(\pi) = 0,\]
are given by

A

\(\lambda = (n\pi)^2, \quad n = 1,2,3,...\)

B

\(\lambda = n^2, \quad n = 1,2,3,...\)

C

\(\lambda = k_n^2\), where \(k_n, n = 1,2,3,...\) are the roots of \(k - \tan(k\pi) = 0\)

D

\(\lambda = k_n^2\), where \(k_n, n = 1,2,3,...\) are the roots of \(k + \tan(k\pi) = 0\)

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13

2022 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2022

If eigenfunctions corresponding to distinct eigenvalues \(\lambda\) of the Sturm-Liouville problem \[ \frac{d^2y}{dx^2} - 3\frac{dy}{dx} = \lambda y, \quad 0 < x < \pi, \quad y(0) = y(\pi) = 0 \] are orthogonal with respect to the weight function \(w(x)\), then \(w(x)\) is

A

\(e^{-3x}\)

B

\(e^{-2x}\)

C

\(e^{2x}\)

D

\(e^{3x}\)

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14

2022 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2022

The Bessel functions \(J_\alpha(x)\), \(x > 0\), \(\alpha \in \mathbb{R}\) satisfy \(J_{\alpha-1}(x) + J_{\alpha+1}(x) = \frac{2\alpha}{x} J_\alpha(x)\). Then, the value of \((\pi J_{\frac{3}{2}}(\pi))^2\) is __________.

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15

2023 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2023

Let \(\phi\) and \(\psi\) be two linearly independent solutions of the ordinary differential equation
\[ y'' + (2 - \cos x) y = 0, \quad x \in \mathbb{R}. \]
Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha < \beta\), \(\phi(\alpha) = \phi(\beta) = 0\) and \(\phi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Consider the following statements:
\(P:\) \(\phi'(\alpha)\phi'(\beta) > 0\).
\(Q:\) \(\phi(x)\psi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Then

A

\(P\) is TRUE and \(Q\) is FALSE

B

\(P\) is FALSE and \(Q\) is TRUE

C

both \(P\) and \(Q\) are FALSE

D

both \(P\) and \(Q\) are TRUE

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16

2023 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2023

For every \(k \in \mathbb{N} \cup \{0\}\), let \(y_k(x)\) be a polynomial of degree \(k\) with \(y_k(1) = 5\). Further, let \(y_k(x)\) satisfy the Legendre equation \[ (1 - x^2)y'' - 2xy' + k(k+1)y = 0. \] If \[ \frac{1}{2} \int_{-1}^{1} \sum_{k=1}^{n} (y_k(x) - y_{k-1}(x))^2 \, dx - \int_{-1}^{1} \sum_{k=1}^{n} (y_k(x))^2 \, dx = 24, \] for some positive integer \(n\), then the value of \(n\) is ________.

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17

2024 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2024

Which of the following is/are eigenvalue(s) of the Sturm-Liouville problem \(y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \quad y(0) = y'(0), \quad y(\pi) = y'(\pi)?\)

A

\(\lambda = 1\)

B

\(\lambda = 2\)

C

\(\lambda = 3\)

D

\(\lambda = 4\)

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18

2024 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2024

The boundary value problem \(x^2y'' - 2xy' + 2y = 0\), \(1 \le x \le 2\), \(y(1) - y'(1) = 1\), \(y(2) - ky'(2) = 4\), has infinitely many distinct solutions when \(k\) is equal to ______ (round off to TWO decimal places)

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19

2025 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2025

Let \(y = P_n(x)\) be the unique polynomial of degree \(n\) satisfying the Legendre differential equation
\((1 - x^2)y'' - 2xy' + n(n + 1)y = 0\) and \(y(1) = 1\).
Then, the value of \(P'_{11}(1)\) is equal to ____ (in integer)

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20

2007 · Mathematics · Ordinary Differential Equations · Sturm-Liouville Problems and Special Functions

Mathematics (MA) 2007

Let \(P_n(x)\) be the Legendre polynomial of degree \(n\) and let \(P_{n+1}'(0) = -\frac{m}{m+1} P_{n-1}'(0), m = 1, 2, ... .\) If \(P_n'(0) = -\frac{5}{16}\), then \(\int_{-1}^1 P_n^2(x) dx =\)

A

\(\frac{2}{13}\)

B

\(\frac{2}{9}\)

C

\(\frac{5}{16}\)

D

\(\frac{2}{5}\)

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