Chapter 3: Q 3.6-5E (page 130)

Show that when the improved Euler’s 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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Show that when the improved Euler’s 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 .

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Show that when the improved Euler’s 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

Discrete Mathematics

Probability and Statistics

Logic and Functions

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Show that when the improved Euler’s 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 $\frac{1}{3} {(1 + 4 h + 8 h^{2})}^{\frac{1}{2 h}}$ .