Q 3.6-5E Show that when the improved Eule... [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 3.6-5E (page 130)

Show that when the improved Euler鈥檚 method is used to approximate the solution of the initial value problem $y' = 4 y, y (0) = \frac{1}{3}$ , at $x = \frac{1}{2}$ , then the approximation with step size his $\frac{1}{3} {(1 + 4 h + 8 h^{2})}^{\frac{1}{2 h}}$ .

Short Answer

Expert verified

Proved

Step by step solution

Find the value of yn

Here f(x, y) = 4y. Apply the formula

$y_{n + 1} = y_{n} + \frac{h}{2} [f (x_{n}, y_{n}) + f (x_{n} + h, y_{n} + hf (x_{n}, y_{n})] = y_{n} + \frac{h}{2} [4 y_{n}) + f (x_{n} + h, y_{n} + 4 {hy}_{n})] = y_{n} + \frac{h}{2} [4 y_{n}) + 4 (y_{n} + 4 {hy}_{n})] = y_{n} + \frac{h}{2} [8 y_{n}) + 16 h y_{n})] = y_{n} + 4 {hy}_{n} + 8 h^{2} y_{n} y_{n + 1} = y_{n} (1 + 4 h + 8 h^{2})$

Evaluate the approximation value for x=12 .

Since $y (0) = \frac{1}{3}$

$y_{1} = \frac{1}{3} (1 + 4 h + 8 h^{2}) y_{2} = \frac{1}{3} (1 + 4 h + 8 h^{2})^{2} y_{3} = \frac{1}{3} (1 + 4 h + 8 h^{2})^{3} . . . y_{n} = \frac{1}{3} (1 + 4 h + 8 h^{2})^{n}$

Since $x_{o} = 0, x = \frac{1}{2}$ then

$\frac{1}{2} = x_{o} + nh = nh n = \frac{1}{2 h}$

Then

$y_{n} = \frac{1}{3} (1 + 4 h + 8 h^{2})^{\frac{1}{2 h}}$

Hence it is proved that role="math" localid="1664233218531" $y_{n} = \frac{1}{3} (1 + 4 h + 8 h^{2})^{\frac{1}{2 h}}$

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91影视

Show that when the improved Euler鈥檚 method is used to approximate the solution of the initial value problem $y' = 4 y, y (0) = \frac{1}{3}$ , at $x = \frac{1}{2}$ , then the approximation with step size his $\frac{1}{3} {(1 + 4 h + 8 h^{2})}^{\frac{1}{2 h}}$ .

Short Answer

Step by step solution

Find the value of yn

Evaluate the approximation value for x=12 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Show that when the improved Euler鈥檚 method is used to approximate the solution of the initial value problem y'=4y,y(0)=13, atx=12 , then the approximation with step size his13(1+4h+8h2)12h .

Short Answer

Step by step solution

Find the value of yn

Evaluate the approximation value for x=12 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Geometry

Discrete Mathematics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Show that when the improved Euler鈥檚 method is used to approximate the solution of the initial value problem $y' = 4 y, y (0) = \frac{1}{3}$ , at $x = \frac{1}{2}$ , then the approximation with step size his $\frac{1}{3} {(1 + 4 h + 8 h^{2})}^{\frac{1}{2 h}}$ .