Numerical solution of initial value problems for ordinary differential equations - Numerical Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2008

Consider the initial value problem \(\frac{dy}{dx} = f(x, y), \; y(x_0) = y_0\). The aim is to compute the value of \(y_1 = y(x_0 + h)\), where \(x_0 = x_0 + h \; (h > 0)\). At \(x = x_1\), if the value of \(y_1\) is equated to the corresponding value of the straight line passing through \((x_0, y_0)\) and having the slope equal to the slope of the curve \(y(x)\) at \(x = x_0\), then the method is called

A

Euler's method

B

Improved Euler's method

C

Backward Euler's method

D

Taylor series method of order 2

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2

2008 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2008

If \(y_{i+1} = y_i + h \phi(f, x_i, y_i, h), i = 1, 2, \cdots\), where \(\phi(f, x, y, h) = a f(x, y) + b f(x+h, y+h f(x, y))\), is a second order accurate scheme to solve the initial value problem \(\frac{dy}{dx} = f(x, y), y(x_0) = y_0\), then \(a\) and \(b\), respectively, are

A

\(\frac{h}{2}, \frac{h}{2}\)

B

\(1, -1\)

C

\(\frac{1}{2}, \frac{1}{2}\)

D

\(h, -h\)

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3

2011 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2011

Let \(y\) be the solution of the initial value problem \[\frac{dy}{dx} = (y^2 + x); \quad y(0) = 1.\] Using Taylor series method of order 2 with the step size \(h=0.1\), the approximate value of \(y(0.1)\) is

A

1.315

B

1.415

C

1.115

D

1.215

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4

2012 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2012

Using Euler's method taking step size \( = 0.1 \), the approximate value of \( y \) obtained corresponding to \( x = 0.2 \) for the initial value problem \( \frac{dy}{dx} = x^{2} + y^{2} \) and \( y(0) = 1 \), is

A

1.322

B

1.122

C

1.222

D

1.110

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5

2014 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2014

The fourth order Runge-Kutta method given by
\(u_{j+1}=u_{j}+\frac{h}{6}\left[K_{1}+2 K_{2}+2 K_{3}+K_{4}\right], \quad j=0,1,2, \ldots,\)
is used to solve the initial value problem \(\frac{d u}{d t}=u, \quad u(0)=\alpha\).
If \(u(1)=1\) is obtained by taking the step size \(h=1\), then the value of \(K_{4}\) is ______________

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6

2014 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2014

Let \(y(x)\) be the solution to the initial value problem \(\frac{dy}{dx} = \sqrt{y + 2x}\) subject to \(y(1.2) = 2\). Using the Euler method with the step size \(h = 0.05\), the approximate value of \(y(1.3)\), correct to two decimal places, is ______________

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7

2019 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2019

If the differential equation \[\frac{dy}{dx} = \sqrt{x^2 + y^2}, \ y(1) = 2\] is solved using the Euler's method with step-size \(h = 0.1\), then \(y(1.2)\) is equal to ______ (round off to 2 places of decimal).

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8

2021 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2021

The initial value problem \(\frac{dy}{dt} = f(t, y),\ t > 0,\ y(0) = 1\), where \(f(t, y) = -10 y\), is solved by the following Euler method \(y_{n+1} = y_n + h f(t_n, y_n),\ n \ge 0\), with step-size \(h\). Then \(y_n \to 0\) as \(n \to \infty\), provided

A

\(0 < h < 0.2\)

B

\(0.3 < h < 0.4\)

C

\(0.4 < h < 0.5\)

D

\(0.5 < h < 0.55\)

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9

2022 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2022

The initial value problem \(\frac{dy}{dx} = f(x, y), y(x_0) = y_0\) is solved by using the following second order Runge-Kutta method:
\(K_1 = h f(x_i, y_i)\)
\(K_2 = h f(x_i + \alpha h, y_i + \beta K_1)\)
\(y_{i+1} = y_i + \frac{1}{4}(K_1 + 3K_2), \; i \geq 0,\)
where \(h\) is the uniform step length between the points \(x_0, x_1, \cdots , x_n\) and \(y_i = y(x_i)\). The value of the product \(\alpha \beta\) is __________ (round off to TWO decimal places).

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10

2024 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2024

For the initial value problem
\[ y' = f(x, y), \quad y(x_0) = y_0, \]
generate approximations \(y_n\) to \(y(x_n)\), \(x_n = x_0 + nh\), for a fixed \(h > 0\) and \(n = 1, 2, 3, \ldots\), using the recursion formula
\[ y_n = y_{n-1} + a k_1 + b k_2, \text{ where} \]
\[ k_1 = h f(x_{n-1}, y_{n-1}) \text{ and } k_2 = h f(x_{n-1} + \alpha h, y_{n-1} + \beta k_1). \]
Which one of the following choices of \(a, b, \alpha, \beta\) for the above recursion formula gives the Runge-Kutta method of order 2 ?

A

\(a = 1, b = 1, \alpha = 0.5, \beta = 0.5\)

B

\(a = 0.5, b = 0.5, \alpha = 2, \beta = 2\)

C

\(a = 0.25, b = 0.75, \alpha = 2/3, \beta = 2/3\)

D

\(a = 0.5, b = 0.5, \alpha = 1, \beta = 2\)

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11

2025 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2025

Let \(y(x)\) be the solution of the initial value problem \(\frac{dy}{dx} = \sin(\pi(x + y))\), \(y(0) = 0\). Using Euler's method, with the step-size \(h = 0.5\), the approximate value of \(y(1.5) + 2y(1)\) is equal to ____ (in integer)

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12

2026 · Mathematics · Numerical Analysis · Numerical solution of initial value problems for ordinary differential equations

Mathematics (MA) 2026

Consider the differential equation \(\frac{dy}{dx} = x + y\) with the initial condition \(y(0) = 1\). Using the modified Euler's method, the second approximation to \(y(h)\), where \(h = 0.05\) (step size), is equal to ______. (rounded off to TWO decimal places)

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