In Problem.apply the successive approximationformula to compute yn(x) for n 4. Then write the expo-nential series for which these approximations are partial sums(perhaps with the first term or two missing; for example,1,2= -2y, y(0) = 4dx

Answered: In Problem. apply the successive…

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

In Problem.
apply the successive approximation
formula to compute yn(x) for n 4. Then write the expo-
nential series for which these approximations are partial sums
(perhaps with the first term or two missing; for example,
1,2
= -2y, y(0) = 4
dx

Expert Solution

Step 1

Given,

$\frac{d y}{d x} = - 2 y, y (0) = 4 .$

Step 2

By using the Picard iteration method:

$\frac{d y}{d x} = f (x, y), y (x_{0}) = y_{0}, . . . (1) y (x) = y_{0} + \int_{x_{0}}^{x} f (t, y (t)) d t, . . . (2)$

From (1) and (2), we obtain

$y_{n} (x) = y_{0} + \int_{x_{0}}^{x} f (t, y_{n - 1} (t)) d t, n = 1, 2, 3 .$

Now,

$\frac{d y}{d x} = - 2 y, y (0) = 4, \Rightarrow f (x, y) = - 2 y, y_{0} = 4, x_{0} = 0 .$

$∴ y_{1} = y_{0} + \int_{x_{0}}^{x} f (t, 4) d t = 4 + \int_{0}^{x} (- 8) d t, = 4 - 8 \int_{0}^{x} d t, = 4 - 8 {[t]}_{0}^{x}, = 4 - 8 x .$

Step 3

$y_{2} = y_{1} + \int_{x_{0}}^{x} f (t, y_{1}) d t = 4 + \int_{0}^{x} - 2 (4 - 8 t) d t, = 4 - 8 \int_{0}^{x} (1 - 2 t) d t, = 4 - 8 {[t - t^{2}]}_{0}^{x}, = 4 - 8 (x - x^{2}) = 4 - 8 x + 8 x^{2} .$

$y_{3} = y_{0} + \int_{x_{0}}^{x} f (t, y_{2}) d t = 4 + \int_{0}^{x} - 2 (4 - 8 t + 8 t^{2}) d t = 4 - 8 \int_{0}^{x} (1 - 2 t + 2 t^{2}) d t = 4 - 8 {[t - t^{2} + \frac{2}{3} t^{3}]}_{0}^{x}, = 4 - 8 x + 8 x^{2} - \frac{16}{3} x^{3} .$