Consider the following initial value problems. Useimproved Euler's method, with the step size h = 0.1 to obtain a four-decimal approximationof the indicated values. Solve this question without using EXCEL.y' = e",y (0) = 0;y(0.5)

Answered: Consider the following initial value…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the following initial value problems. Use
improved Euler's method, with the step size h = 0.1 to obtain a four-decimal approximation
of the indicated values. Solve this question without using EXCEL.
y' = e",
y (0) = 0;
y(0.5)

Expert Solution

Step 1

The given inital value problem is $y' = e^{- y}$ and $y (0) = 0$ .

The value of h is given as 0.1.

Compute the value of y(0.5) as follows.

It is known that the formula for the improved Euler's method is $y_{n + 1} = y_{n} + \frac{h}{2} (f (x_{n}, y_{n}) + f (x_{n + 1}, {\bar{y}}_{n + 1}))$ , where ${\bar{y}}_{n + 1} = y_{n} + h f (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + h$ .

Step 1:

Compute x₁ as follows.

$x_{1} = x_{0} + h = 0 + 0.1 = 0.1$

Obtain the value of ${\bar{y}}_{1}$ as follows.

${\bar{y}}_{1} = y_{0} + 0.1 f (x_{0}, y_{0}) = 0 + 0.1 f (0, 0) = 0 + 0.1 (e^{0}) = 0.1$

compute y₁ as follows.

$y_{1} = y_{0} + \frac{h}{2} (f (x_{0}, y_{0}) + f (x_{1}, {\bar{y}}_{1})) = 0 + \frac{0.1}{2} (f (0, 0) + f (0.1, 0.1) = 0 + 0.05 (1 + e^{0.1}) = 0.095241870901798$

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