Autonomous Systems and Stability - Ordinary Differential Equations - Mathematics Previous Year Questions

1

2008 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2008

Let \(a, b \in \mathbb{R}\). Let \(y = (y_1, y_2)^T\) be a solution of the system of equations \[ y_1' = y_2, \ y_2' = a y_1 + b y_2. \] Every solution \(y(x) \to 0\) as \(x \to \infty\) if

A

a < 0, b < 0

B

a < 0, b > 0

C

a > 0, b > 0

D

a > 0, b < 0

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2

2008 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2008

In the closed system of a simple harmonic motion of a pendulum, let \(H\) denote the Hamiltonian and \(E\) be the total energy. Then

A

\(H\) is a constant and \(H = E\)

B

\(H\) is a constant but \(H \neq E\)

C

\(H\) is not constant but \(H = E\)

D

\(H\) is not constant and \(H \neq E\)

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3

2009 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2009

A simple pendulum, consisting of a bob of mass \( m \) connected with a string of length \( a \), is oscillating in a vertical plane. If the string is making an angle \( \theta \) with the vertical, then the expression for the Lagrangian is given as

A

\(ma^2 \left( \dot{\theta}^2 - \frac{2g}{a} \sin^2 \left( \frac{\theta}{2} \right) \right) \)

B

\(2mga \sin^2 \left( \frac{\theta}{2} \right) \)

C

\(ma^2 \left( \dot{\theta}^2 - \frac{2g}{a} \sin^2 \left( \frac{\theta}{2} \right) \right) \)

D

\(\frac{ma}{2} \left( \dot{\theta}^2 - \frac{2g}{a} \cos \theta \right) \)

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4

2010 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2010

Let \( H, T \) and \( V \) denote the Hamiltonian, the kinetic energy and the potential energy respectively of a mechanical system at time \( t \). If \( H \) contains \( t \) explicitly, then \( \frac{\partial H}{\partial t} \) is equal to

A

\(\frac{\partial T}{\partial t} + \frac{\partial V}{\partial t} \)

B

\(\frac{\partial T}{\partial t} - \frac{\partial V}{\partial t} \)

C

\(\frac{\partial V}{\partial t} - \frac{\partial T}{\partial t} \)

D

\(-\frac{\partial V}{\partial t} - \frac{\partial T}{\partial t} \)

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5

2021 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2021

The critical point of the differential equation \(\frac{d^2 y}{dt^2} + 2 \alpha \frac{dy}{dt} + \beta^2 y = 0,\ \alpha > \beta > 0\), is a

A

node and is asymptotically stable

B

spiral point and is asymptotically stable

C

node and is unstable

D

saddle point and is unstable

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6

2025 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2025

Consider the system of ordinary differential equations \( \frac{dX}{dt} = MX \), where \( M \) is a \( 6 \times 6 \) skew-symmetric matrix with entries in \( \mathbb{R} \). Then, for this system, the origin is a stable critical point for

A

any such matrix \(M \)

B

only such matrices \(M \) whose rank is 2

C

only such matrices \(M \) whose rank is 4

D

only such matrices \(M \) whose rank is 6

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7

2007 · Mathematics · Ordinary Differential Equations · Autonomous Systems and Stability

Mathematics (MA) 2007

Let \(Y(x) = (y_1(x), y_2(x))\) and let \(A = \begin{bmatrix} -3 & 1 \\ k & -1 \end{bmatrix}\).
Further, let \(S\) be the set of values of \(k\) for which all the solutions of the system of equations \(Y'(x) = A Y(x)\) tend to zero as \(x \to \infty\). Then \(S\) is given by

A

\(\{k : k \leq -1\}\)

B

\(\{k : k \leq 3\}\)

C

\(\{k : k < -1\}\)

D

\(\{k : k < 3\}\)

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