Answered: A hand-held calculator will suffice for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A hand-held calculator will suffice for the Problem. The problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2)in three ways: (a) by the Euler method with two steps of size h = 0.1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge–Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2).

x' = x -2y, x(0) = 0,

y' = 2x + y, y(0) = 4;

x(t) = -4e^t sin 2t, y(t) = 4e^tcos 2t

Expert Solution

Step 1

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Step 2

Given an initial value problem are

$x' = x - 2 y, x (0) = 0 y' = 2 x + y, y (0) = 4$

and its exact solution are given

$x (t) = - 4 e^{t} \sin (2 t) y (t) = 4 e^{t} \cos (2 t)$

and (steps of size) h= 0.1

Step 3

Using the Euler method

$x_{n + 1} = x_{n} + h G (t_{n}, x_{n}, y_{n}) y_{n + 1} = y_{n} + h H (t_{n}, x_{n}, y_{n})$

and $t_{n + 1} = t_{n} + h$

Step 4

Now

$x' = x - 2 y \Rightarrow G (t_{n}, x_{n}, y_{n}) = x_{n} - 2 y_{n} y' = 2 x + y \Rightarrow H (t_{n}, x_{n}, y_{n}) = 2 x_{n} + y_{n}$

Then

Iteration First: Put n=0

$x_{1} = x_{0} + h G (t_{0}, x_{0}, y_{0}) x_{1} = x_{0} + h (x_{0} - 2 y_{0})$

Using $x (0) = 0, y (0) = 4$

$x_{1} = 0 + 0.1 (0 - 2 \times 4) x_{1} = - 0.8$

and

$y_{1} = y_{0} + h H (t_{0}, x_{0}, y_{0}) y_{1} = y_{0} + h (2 x_{0} + y_{0})$

Using $x (0) = 0, y (0) = 4$

$y_{1} = 4 + 0.1 (2 \times 0 + 4) y_{1} = 4.4$

and $t_{1} = t_{0} + h = 0 + 0.1 = 0.1$

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