Q 3.7-4E Determine the recursive formulas... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 3: Q 3.7-4E (page 139)

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .

Short Answer

Expert verified

$y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

Step by step solution

Find the value of f2(x,y)

Here $y' = x^{2} + y, y (0) = 0$

Apply the chain rule.

$f_{2} (x, y) = \frac{鈭� f}{鈭� x} (x, y) + \frac{鈭� f}{鈭� y} (x, y) f (x, y)$

Since ${f(x,y) =x}^{2} + y$

$\frac{鈭� f}{鈭� x} (x, y) = 2 x \frac{鈭� f}{鈭� y} (x, y) = 1$

So, the equation is $f_{2} (x, y) = 2 x + x^{2} + y$

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the same procedure as step 1

$f_{3} (x, y) = 2 + 2 x + x^{2} + y f_{4} (x, y) = 2 + 2 x + x + y$

Apply the recursive formulas for order 4

The recursive formula is

$x_{n + 1} = x_{n} + h y_{n + 1} = y_{n} + hf (x_{n} + y_{n}) + \frac{h^{2}}{2!} f_{2} (x_{n} + y_{n}) + . . . . . \frac{h^{p}}{p!} f_{p} (x_{n} + y_{n})$

$x_{n + 1} = x_{n} + h y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

Where starting points are $x_{o} = 0, y_{0} = 0$ .

Hence the solution is

role="math" localid="1664317215716" $y_{n + 1} = y_{n} + h ({x_{n}}^{2} + y_{n}) + \frac{h^{2}}{2} (2 x + x^{2} + y) + \frac{h^{3}}{6} (2 + 2 x + x^{2} + y) + \frac{h^{4}}{24} (2 + 2 x + x^{2} + y)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .

Short Answer

Step by step solution

Find the value of f2(x,y)

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the recursive formulas for order 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Probability and Statistics

Calculus

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem y'=x2+y,y(0)=0.

Short Answer

Step by step solution

Find the value of f2(x,y)

Evaluate the values of f3(x,y) and f4 (x,y)

Apply the recursive formulas for order 4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Probability and Statistics

Calculus

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $y' = x^{2} + y, y (0) = 0$ .