Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the value of f2(x,y)

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

Apply the chain rule.

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

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

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

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

Step 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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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

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

Step by Step Solution

Step 1: Find the value of f2(x,y)

Step 2: Apply the recursive formulas for order 2

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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

Step by Step Solution

Step 1: Find the value of f2(x,y)

Step 2: Apply the recursive formulas for order 2

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