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

Use Euler’s method (4) with h = 0.1 to approximate the solution to the initial value problem $y' = - 20 y, y (0) = 1$ , on the interval $0 ⩽ x ⩽ 1$ (that is, at x = 0, 0.1, . . . , 1.0).Compare your answers with the actual solution y = e^-20x. What went wrong? Next, try the step size h = 0.025 and also h = 0.2. What conclusions can you draw concerning the choice of step size?

x_n	y_n(h = 0.2)	y_n(h = 0.1)	y_n(h = 0.025)
0.1		-1	0.06250
0.2	-3	1	0.00391
0.3		-1	0.00024
0.4	9	1	0.00002
0.5		-1	0.00000
0.6	-27	1	0.00000
0.7		-1	0.00000
0.8	81	1	0.00000
0.9		-1	0.00000
1.0	-243	1	0.00000

The result is for h=0.1 the values are 1 and -1. The values for h=0.2 the results are increasing and decreasing with different signs. For h=0.025 is the best approximation result.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the equation of approximation value

Here f(x,y)=-20y then

$y_{n} = y_{n - 1} + h (- 20 y_{n - 1}) = (1 - 20 h) y_{n - 1} . . . = (1 - 20 h)^{n} y_{o} (y_{0} = 1) (1 - 20 h)^{n}$

Step 2: Solve for x and y

Apply initial points $x_{n} = x_{o} + nh, x_{o} = 0$ for

$h = 0.1 x_{n} = 0.1 n, y_{n} = (- 1)^{n}, n = 1, 2, . . . . . 10 h = 0.025 x_{n} = 0.025 n, y_{n} = (0 . 5)^{n}, n = 1, 2, . . . 40 h = 0.2 x_{n} = 0.2 n, y_{n} = (- 3)^{n}, n = 1, 2, . . . 5$

Where $n = 1, 2, 3, . . . . \frac{1}{h}$

Step 3: Determine the all other values for h=0.1 and substituting the value of n

$(x_{1} = 0.1, y_{1} = - 1) (x_{2} = 0.2, y_{2} = 1) (x_{3} = 0.3, y_{3} = - 1) (x_{4} = 0.4, y_{4} = 1) (x_{5} = 0.5, y_{5} = - 1) (x_{6} = 0.6, y_{6} = 1) (x_{7} = 0.7, y_{7} = - 1) (x_{8} = 0.8, y_{8} = 1) (x_{9} = 0.9, y_{9} = - 1) (x_{10} = 1, y_{10} = 1)$

Step 4: Evaluate the all other values for h=0.025 and substituting the value of n

$(x_{1} = 0.1, y_{1} = 0 . 0625) (x_{2} = 0.2, y_{2} = 0 . 003906) (x_{3} = 0.3, y_{3} = 0 . 000244) (x_{4} = 0.4, y_{4} = 0 . 000015) (x_{5} = 0.5, y_{5} = 0 . 000001) (x_{6} = 0.6, y_{6} = 0) (x_{7} = 0.7, y_{7} = 0) (x_{8} = 0.8, y_{8} = 0) (x_{9} = 0.9, y_{9} = 0) (x_{10} = 1, y_{10} = 0)$

Step 5: Find the all-other values for h=0.2 and substituting the value of n

$(x_{2} = 0.2, y_{2} = - 3) (x_{4} = 0.4, y_{4} = 9) (x_{6} = 0.6, y_{6} = - 27) (x_{8} = 0.8, y_{8} = 81) (x_{10} = 1, y_{10} = - 243)$

Hence the solution is

x_n	y_n(h = 0.2)	y_n(h = 0.1)	y_n(h = 0.025)
0.1		-1	0.06250
0.2	-3	1	0.00391
0.3		-1	0.00024
0.4	9	1	0.00002
0.5		-1	0.00000
0.6	-27	1	0.00000
0.7		-1	0.00000
0.8	81	1	0.00000
0.9		-1	0.00000
1.0	-243	1	0.00000

The result is for h=0.1 the values are 1 and -1. The values for h=0.2 the results are increasing and decreasing with different signs. For h=0.025 is the best approximation result.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Step by Step Solution

Step 1: Find the equation of approximation value

Step 2: Solve for x and y

Step 3: Determine the all other values for h=0.1 and substituting the value of n

Step 4: Evaluate the all other values for h=0.025 and substituting the value of n

Step 5: Find the all-other values for h=0.2 and substituting the value of n

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Find the equation of approximation value

Step 2: Solve for x and y

Step 3: Determine the all other values for h=0.1 and substituting the value of n

Step 4: Evaluate the all other values for h=0.025 and substituting the value of n

Step 5: Find the all-other values for h=0.2 and substituting the value of n

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