Q 3.7-2E Determine the recursive formulas... [FREE SOLUTION]

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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 3.7-2E (page 139)

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem $y' = xy - y^{2}, y (0) = - 1$ .

Short Answer

Expert verified

$y_{n + 1} = y_{n} + h (x_{n} y_{n} + {y_{n}}^{2}) - \frac{h^{2}}{2} (y_{n} + (x_{n} - 2 y_{n}) (x_{n} y_{n} - {y_{n}}^{2}))$

Step by step solution

Find the value of f2(x,y)

Here $y' = xy - y^{2}, y (0) = - 1$

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 $f (x, y) = xy - y^{2}$

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

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

Apply the recursive formulas for order 2

The recursive formula is

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

for order p = 2 then

$x_{n + 1} = x_{n} + h y_{n + 1} = y_{n} + h (x_{n} y_{n} + {y_{n}}^{2}) - \frac{h^{2}}{2} (y_{n} + (x_{n} - 2 y_{n}) (x_{n} y_{n} - {y_{n}}^{2}))$

Where starting points are $x_{o} = 0, y_{0} = - 1$ .

Hence, the solution is role="math" localid="1664315352749" $y_{n + 1} = y_{n} + h (x_{n} y_{n} + {y_{n}}^{2}) - \frac{h^{2}}{2} (y_{n} + (x_{n} - 2 y_{n}) (x_{n} y_{n} - {y_{n}}^{2}))$

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91影视

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem $y' = xy - y^{2}, y (0) = - 1$ .

Short Answer

Step by step solution

Find the value of f2(x,y)

Apply the recursive formulas for order 2

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91影视

Determine the recursive formulas for the Taylor method of order 2 for the initial value problemy'=xy-y2,y(0)=-1.

Short Answer

Step by step solution

Find the value of f2(x,y)

Apply the recursive formulas for order 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem $y' = xy - y^{2}, y (0) = - 1$ .