Laplace Transforms and Series Solutions - Ordinary Differential Equations - Mathematics Previous Year Questions

1

2011 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2011

Let \( y \) be the solution of the initial value problem \( \frac{d^2 y}{dx^2} + y = 6 \cos 2x, \quad y(0) = 3, y'(0) = 1. \) Let the Laplace transform of \( y \) be \( F(s). \) Then, the value of \( F(1) \) is

A

17/5

B

13/5

C

11/5

D

9/5

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2

2012 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2012

The expression \(\frac{1}{D_x^2-D_y^2}\sin(x-y)\) is equal to

A

\(-\frac{x}{2}\cos(x-y)\)

B

\(-\frac{x}{2}\sin(x-y)+\cos(x-y)\)

C

\(-\frac{x}{2}\cos(x-y)+\sin(x-y)\)

D

\(\frac{3x}{2}\sin(x-y)\)

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3

2012 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2012

The function \(\phi(x)\) satisfying the integral equation \[\int_0^x e^{x-\xi}\phi(\xi)\,d\xi=\frac{x^2}{2}\] is

A

\(\frac{x^2}{2}\)

B

\(x+\frac{x^2}{2}\)

C

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

D

\(1+\frac{x^2}{2}\)

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4

2012 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2012

The solution of the initial value problem \(y''+2y'+10y=6\delta(t),\quad y(0)=0,\ y'(0)=0,\) where \(\delta(t)\) denotes the Dirac-delta function, is

A

\(2e^{-t}\sin3t\)

B

\(6e^{-t}\sin3t\)

C

\(2e^{-t}\sin3t\)

D

\(6e^{-t}\sin3t\)

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5

2012 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2012

The resolvent kernel \( R(x,t;\lambda) \) for this integral equation is

A

\(\frac{xe^{t}}{1-\lambda} \)

B

\(\frac{\lambda x e^{t}}{1+\lambda} \)

C

\(\frac{xe^{t}}{1+\lambda^{2}} \)

D

\(\frac{xe^{t}}{1-\lambda^{2}} \)

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6

2014 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2014

The solution to the initial value problem \(\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + 5y = 3e^{-t} \sin t\), \(y(0) = 0\) and \(\frac{dy}{dt}(0) = 3\), is

A

\(y(t) = e^t (\sin t + \sin 2t)\)

B

\(y(t) = e^{-t} (\sin t + \sin 2t)\)

C

\(y(t) = 3e^t \sin t\)

D

\(y(t) = 3e^{-t} \sin t\)

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7

2014 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2014

The solution to the integral equation \(\varphi(x) = x + \int_{0}^{x} \sin(x - \xi) \varphi(\xi) d\xi\) is

A

\(x^2 + \frac{x^3}{3}\)

B

\(x - \frac{x^3}{3!}\)

C

\(x + \frac{x^3}{3!}\)

D

\(x^2 - \frac{x^3}{3!}\)

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8

2014 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2014

The general solution to the ordinary differential equation \(x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(4x^2 - \frac{9}{25}\right) y = 0\) in terms of Bessel’s functions, \(J_\nu(x)\), is

A

\(y(x) = c_1 J_{-3/5}(2x) + c_2 J_{-2/5}(2x)\)

B

\(y(x) = c_1 J_{3/10}(x) + c_2 J_{-3/10}(x)\)

C

\(y(x) = c_1 J_{3/5}(x) + c_2 J_{-3/5}(x)\)

D

\(y(x) = c_1 J_{3/10}(2x) + c_2 J_{-3/10}(2x)\)

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9

2014 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2014

The inverse Laplace transform of \(\frac{2s^2 - 4}{(s-3)(s^2 - s - 2)}\) is

A

\((1+e^{-t} + \frac{7}{2} e^{-3t})\)

B

\((\frac{e^t}{3} + t e^{-t} + 2t)\)

C

\((\frac{7}{2} e^{3t} - \frac{e^{-t}}{6} - \frac{4}{3} e^{-2t})\)

D

\((\frac{7}{2} e^{-3t} - \frac{e^{t}}{6} - \frac{4}{3} e^{2t})\)

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10

2015 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2015

Let \(y(t)\) be a continuous function on \([0, \infty)\) whose Laplace transform exists. If \(y(t)\) satisfies \[\int_0^t (1 - \cos(t - \tau)) y(\tau) d\tau = t^4,\] then \(y(1)\) is equal to ________

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11

2018 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2018

If the Laplace transform of \(y(t)\) is given by \(Y(s) = L(y(t)) = \frac{5}{2(s-1)} - \frac{2}{s-2} + \frac{1}{2(s-3)}\), then \(y(0) + y'(0) = \) ________.

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12

2019 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2019

Consider the differential equation \(\frac{d^2 y}{dt^2} + 2 \frac{dy}{dt} + y = 0, t > 0, y(0+) = 1, \left( \frac{dy}{dt} \right)_{t=0+} = 0\).
If \(Y(s)\) is the Laplace transform of \(y(t)\), then the value of \(Y(1)\) is ______ (round off to 2 places of decimal).
(Here, the inverse trigonometric functions assume principal values only)

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13

2021 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2021

Let \(y(t)\) be the solution of the initial value problem \(\frac{d^2 y}{dt^2} + a \frac{dy}{dt} + b y = f(t),\ a > 0,\ b > 0,\ a \neq b,\ a^2 - 4b = 0,\ y(0) = 0,\ \frac{dy}{dt}(0) = 0\), obtained by the method of Laplace transform. Then

A

\(y(t) = \int_0^t \tau e^{\frac{-a\tau}{2}} f(t - \tau) d\tau\)

B

\(y(t) = \int_0^t e^{\frac{-a\tau}{2}} f(t - \tau) d\tau\)

C

\(y(t) = \int_0^t \tau e^{\frac{-b\tau}{2}} f(t - \tau) d\tau\)

D

\(y(t) = \int_0^t e^{\frac{-b\tau}{2}} f(t - \tau) d\tau\)

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14

2021 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2021

If \(y = \sum_{k=0}^{\infty} a_k x^k\), \((a_0 \neq 0)\) is the power series solution of the differential equation \(\frac{d^2 y}{dx^2} - 24 x^2 y = 0\), then \(\frac{a_4}{a_0} = \) ________.

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15

2023 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2023

Let \( y(t) \) be the solution of the initial value problem
\( y'' + 4y = \begin{cases} t, & 0 \le t \le 2, \\ 2, & 2 < t < \infty, \end{cases} \) and \( y(0) = y'(0) = 0 \).
If \( \alpha = y\left(\frac{\pi}{2}\right) \), then the value of \( \frac{4}{\pi} \alpha \) is __________ (rounded off to 2 decimal places).

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16

2025 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2025

Given that the Laplace transforms of \( J_0(x) \), \( J_0'(x) \) and \( J_0''(x) \) exist, where \( J_0(x) \) is the Bessel function. Let \( Y = Y(s) \) be the Laplace transform of the Bessel function \( J_0(x) \). Then, which one of the following is TRUE?

A

\(\frac{dY}{ds} + \frac{2sY}{s^2+1} = 0, \quad s > 0 \)

B

\(\frac{dY}{ds} - \frac{2sY}{s^2+1} = 0, \quad s > 0 \)

C

\(\frac{dY}{ds} - \frac{sY}{s^2+1} = 0, \quad s > 0 \)

D

\(\frac{dY}{ds} + \frac{sY}{s^2+1} = 0, \quad s > 0 \)

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17

2007 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2007

If \(F(s) = \tan^{-1}(s) + k\) is the Laplace transform of some function \(f(t), \; t \geq 0\), then \(k =\)

A

\(-\pi\)

B

\(-\frac{\pi}{2}\)

C

0

D

\(\frac{\pi}{2}\)

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18

2007 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2007

Let \( J_0(x) \) and \( J_1(x) \) be the Bessel functions of the first kind of orders zero and one, respectively. If \( \mathcal{L}(J_0(t)) = \frac{1}{\sqrt{s^2 + 1}} \), then \( \mathcal{L}(J_1(t)) = \)

A

\(\frac{s}{\sqrt{s^2 + 1}} \)

B

\(\frac{1}{\sqrt{s^2 + 1}} - 1 \)

C

\(1 - \frac{s}{\sqrt{s^2 + 1}} \)

D

\(\frac{s}{\sqrt{s^2 + 1}} - 1 \)

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19

2007 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2007

For \( n \geq 2 \), the coefficients \( c_n \) will satisfy the relation

A

\(n^2 c_n - c_{n-2} = 0 \)

B

\(n^2 c_n + c_{n-2} = 0 \)

C

\(c_n - n^2 c_{n-2} = 0 \)

D

\(c_n + n^2 c_{n-2} = 0 \)

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20

2026 · Mathematics · Ordinary Differential Equations · Laplace Transforms and Series Solutions

Mathematics (MA) 2026

Let \( \delta(t) \) be the unit impulse function defined by \( \delta(x - x_0) = 0 \) when \( x \neq x_0 \) and \[ \int_{-\infty}^{\infty} \delta(x - x_0) \, dx = 1. \] Consider the following initial value problem \[ 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} + 2y = \delta(x - 5) \] with \( y(0) = 0 \) and \( \frac{dy}{dx}(0) = 0 \). Which one of the following is TRUE?

A

\(y(10) = \frac{10}{\sqrt{15}} e^{-5/4} \sin\left( \frac{\sqrt{15}}{4} \right) \)

B

\(y(10) = \frac{2}{\sqrt{15}} e^{-5/4} \sin\left( \frac{5\sqrt{15}}{4} \right) \)

C

\(y(10) = \frac{10}{\sqrt{15}} e^{5/4} \sin\left( \frac{\sqrt{15}}{4} \right) \)

D

\(y(10) = \frac{2}{\sqrt{15}} e^{5/4} \sin\left( \frac{5\sqrt{15}}{4} \right) \)

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