Chapter 3: Q 3.6-1E (page 129)

Show that when Euler’s method is used to approximate the solution of the initial value problem $y' = 5 y$ y(0) = 1 , at x= 1, then the approximation with step size his ${(1 + 5)}^{\frac{1}{h}}$ .

Short Answer

Expert verified

Proved

Step by step solution

Apply Euler’s formula

Here $y' = 5 y, y (0) = 1$

Apply the Euler’s formula

$y_{n + 1} = y_{n} + hf (x_{n}, y_{n})$

In our case of differential equation our equation is

$y_{n + 1} = y_{n} + h 5 y_{n}$

The interval is $\frac{b - a}{n} = \frac{1 - 0}{n} = \frac{1}{n} = h$

Substitute in the Euler’s formula

Now

$y_{1} = 1 (1 + 5 h) y_{2} = 1 + 5 h (1 + 5 h) - (1 + 5 h)^{2} . . . y_{n} = (1 + 5 h)^{n} y_{n} = (1 + 5 h)^{\frac{1}{h}}$

Hence, it is proved that $y_{n} {= (1 + 5h)}^{\frac{1}{h}}$

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Show that when Euler’s method is used to approximate the solution of the initial value problem $y' = 5 y$ y(0) = 1 , at x= 1, then the approximation with step size his ${(1 + 5)}^{\frac{1}{h}}$ .

Short Answer

Step by step solution

Apply Euler’s formula

Substitute in the Euler’s formula

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Show that when Euler’s method is used to approximate the solution of the initial value problem y'=5yy(0) = 1 , at x= 1, then the approximation with step size his(1+5)1h.

Short Answer

Step by step solution

Apply Euler’s formula

Substitute in the Euler’s formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Show that when Euler’s method is used to approximate the solution of the initial value problem $y' = 5 y$ y(0) = 1 , at x= 1, then the approximation with step size his ${(1 + 5)}^{\frac{1}{h}}$ .