Chapter 1: Q 3.6-10E (page 1)

Use the improved Euler’s method subroutine with step size h = 0.1 to approximate the solution to $y' = 4 \cos (x + y), y (0) = 0$ , at the points $x = 0, 0.1, 0.2, . . ., 1.0$ . Use your answers to make a rough sketch of the solution on [0,1].

Short Answer

Expert verified

The required result is

$(x = 0, y = 1), (x = 0.1, y = 0.158447), (x = 0.2, y = 0.4988), (x = 0.3, y = 0.7418),$

$(x = 0.4, y = 0.88777), (x = 0.5, y = 0.95787), (x = 0.6, y = 0.9730), (x = 0.7, y = 0.952329)$

$(x = 0.8, y = 0.906359), (x = 0.9, y = 0.843228), (x = 1, y = 0.768434)$

Step by step solution

Important formula.

For finding the values ofanduse Euler’s formula,

$x_{n + 1} = (x_{n} + h) y_{n + 1} = x_{n} + \frac{h}{2} (F + G)$

find the equation of approximation value

Here $y' = 4 \cos (x + y), y (0) = 0$ , for $0 ⩽ x ⩽ 1$ .

For $h = 0.1, x = 0, y = 1, N = 10$

$F = f (x, y) = 4 \cos (x + y) G = f (x + h, y + h F) = 4 \cos (x + y + 0.1 + 0.4 \cos (x + y))$

solve for x1 and y1

Apply initial points $x_{o} = 0, y_{o} = 1, h = 0.1$

$F (0, 1) = 2.16121 G (0, 1) = 1.00773$

$x_{n + 1} = (x_{n} + h) y_{n + 1} = x_{n} + \frac{h}{2} (F + G)$

$x_{1} = 0 + 0.1 = 0.1 y_{1} = 0 + \frac{0.1}{2} (2.16121 + 1.00773) = 0.158447$

evaluate the value of x2 and u=y2.

Now, the value of F and G

$F (0.1, 0.158447) = 3.86715 G (0.1, 0.158447) = 2.93991 x_{2} = 0.1 + 0.1 = 0.2 y_{2} = 0.4988$

determine the all-other values

Apply the same procedure for all other values and the values are

$(x = 0.2, y = 0.4988) (x = 0.3, y = 0.7418) (x = 0.4, y = 0.88777) (x = 0.5, y = 0.95787) (x = 0.6, y = 0.9730) (x = 0.7, y = 0.952329) (x = 0.8, y = 0.906359) (x = 0.9, y = 0.843228) (x = 1, y = 0.768434)$

plot a graph for all values.

By putting the values of x and y can draw a graph.

Therefore, this is the required result.

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Use the improved Euler’s method subroutine with step size h = 0.1 to approximate the solution to $y' = 4 \cos (x + y), y (0) = 0$ , at the points $x = 0, 0.1, 0.2, . . ., 1.0$ . Use your answers to make a rough sketch of the solution on [0,1].

Short Answer

Step by step solution

Important formula.

find the equation of approximation value

solve for x1 and y1

evaluate the value of x2 and u=y2.

determine the all-other values

plot a graph for all values.

One App. One Place for Learning.

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Use the improved Euler’s method subroutine with step size h = 0.1 to approximate the solution to y'=4cos(x+y),y(0)=0, at the points x=0, 0.1, 0.2,..., 1.0. Use your answers to make a rough sketch of the solution on [0,1].

Short Answer

Step by step solution

Important formula.

find the equation of approximation value

solve for x1 and y1

evaluate the value of x2 and u=y2.

determine the all-other values

plot a graph for all values.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Decision Maths

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the improved Euler’s method subroutine with step size h = 0.1 to approximate the solution to $y' = 4 \cos (x + y), y (0) = 0$ , at the points $x = 0, 0.1, 0.2, . . ., 1.0$ . Use your answers to make a rough sketch of the solution on [0,1].