• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q9E

Expert-verified Found in: Page 259 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

The result is:

${\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[{\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}\mathbf{+}{\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{+}\phantom{\rule{0ex}{0ex}}{\mathbf{hf}}_{\mathbf{1}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{+}{\mathbf{hf}}_{\mathbf{m}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}\right]$

See the step by step solution

## Step 1: Use Euler’s method

Here given Euler’s method of the differential equation:

So, ${\mathbf{y}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{y}}_{\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[\mathbf{f}\mathbf{\left(}{\mathbf{x}}_{\mathbf{n}}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{\right)}\mathbf{+}\mathbf{f}\mathbf{\left(}{\mathbf{x}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{+}\mathbf{hf}\mathbf{\left(}{\mathbf{x}}_{\mathbf{n}}\mathbf{,}{\mathbf{y}}_{\mathbf{n}}\mathbf{\right)}\mathbf{\right)}\right]$

n=0,1,2,…..and ${\mathbf{x}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{n}}\mathbf{+}\mathbf{h}$.

Now,

$\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\mathbf{.}\phantom{\rule{0ex}{0ex}}\mathbf{.}\phantom{\rule{0ex}{0ex}}\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}}\mathbf{\right)}$

## Step 2: Solve for every i

For every I from 1 to m, then;

${\mathbf{t}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\phantom{\rule{0ex}{0ex}}{\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{i}\mathbf{,}\mathbf{n}}\mathbf{+}\frac{\mathbf{h}}{\mathbf{2}}\left[{\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}\mathbf{+}{\mathbf{f}}_{\mathbf{i}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{+}\mathbf{h}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{+}\phantom{\rule{0ex}{0ex}}{\mathbf{hf}}_{\mathbf{1}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{+}{\mathbf{hf}}_{\mathbf{m}}\mathbf{\left(}{\mathbf{t}}_{\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}\mathbf{,}\mathbf{n}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}\mathbf{,}\mathbf{n}}........\mathbf{,}{\mathbf{x}}_{\mathbf{m}\mathbf{,}\mathbf{n}}\mathbf{\right)}\right]$

This is the required result. ### Want to see more solutions like these? 