Chapter 3: Q 3.7-10E (page 139)

Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problem $y' = 1 - xy, y (1) = 1$ at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.

Short Answer

Expert verified

$ϕ (2) = 0.7014$

Step by step solution

Find the values of ki, i = 1, 2, 3, 4

Since $y' = 1 - xy, y (1) = 1$ and $x = x_{0} = 1, y = y_{o} = 1$ and h = 1, M = 10

$k_{1} = h f (x, y) = 1 (1 - 0) = 0 k_{2} = hf (x + \frac{h}{2}, y + \frac{k_{1}}{2}) = - \frac{1}{2} k_{3} = hf (x + \frac{h}{2}, y + \frac{k_{2}}{2}) = - \frac{1}{8} k_{4} = hf (x + h, y + k_{3}) = - \frac{3}{4}$

Find the values of x and y

$x = 1 + 1 = 2 y = 1 + \frac{1}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4}) = 1 + \frac{1}{6} (0 - 2 (\frac{1}{2}) - 2 (\frac{1}{8}) - (\frac{3}{4})) = 0.6667$

Therefore $ϕ (2) = 0.6667$ .

Since $|1 - 0.6667| = 0.3333 > 0.001$

Find the other values

Apply the same procedure for h=0.5, and h=0.25 respectively.

$ϕ (2) = 0.700799$

Thus, $|0.6667 - 0.7008| = 0.0341 > 0.001$

$ϕ (2) = 0.701385$

Hence, $|0.6667 - 0.7014| = 0.0006 < 0.001$

So, $ϕ (2) = 0.7014$ .

Hence the solution is $ϕ (2) = 0.7014$

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Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problem $y' = 1 - xy, y (1) = 1$ at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.

Short Answer

Step by step solution

Find the values of ki, i = 1, 2, 3, 4

Find the values of x and y

Find the other values

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Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problemy'=1-xy,y(1)=1at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.

Short Answer

Step by step solution

Find the values of ki, i = 1, 2, 3, 4

Find the values of x and y

Find the other values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problem $y' = 1 - xy, y (1) = 1$ at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.