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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

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

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

Short Answer

Expert verified

The required result is

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

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

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

Step by step solution

01

Important formula.

For finding the values ofanduse Euler鈥檚 formula,

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

02

find the equation of approximation value

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

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

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

03

solve for x1 and y1

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

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

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

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

04

evaluate the value of x2 and u=y2.

Now, the value of F and G

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

05

determine the all-other values

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

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

06

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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Most popular questions from this chapter

Show that $蠒 (x) = c_{1} \sin x + c_{2} \cos x,$ is a solution to $\frac{d^{2} y}{{dx}^{2}} + y = 0$ for any choice of the constants $c_{1}$ and $c_{2}$ . Thus, $c_{1} \sin x + c_{2} \cos x,$ is a two-parameter family of solutions to the differential equation.

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

$x \frac{d^{2} y}{d x^{2}} + 3 x - \frac{d y}{d x} = e^{3 x}$

(a) Show that $蠒 (x) = x^{2}$ is an explicit solution to $x \frac{dy}{dx} = 2 y$ on the interval $(- 鈭�, 鈭�)$ .

(b) Show that $蠒 (x) = e^{x} - x$ , is an explicit solution to $\frac{dy}{dx} + y^{2} = e^{x} + (1 - 2 x) e^{x} + x^{2} - 1$ on the interval $(- 鈭�, 鈭�)$ .

(c) Show that $蠒 (x) = x^{2} - x^{- 1}$ is an explicit solution to $x^{2} \frac{d^{2} y}{{dx}^{2}} = 2 y$ on the interval $(0, 鈭�)$ .

Lunar Orbit. The motion of a moon moving in a planar orbit about a planet is governed by the equations $\frac{d^{2} x}{{dt}^{2}} = - G \frac{mx}{r^{3}}, \frac{d^{2} y}{{dt}^{2}} = - G \frac{my}{r^{3}}$ where $r = (x^{2} + y^{2})^{\frac{1}{2}}$ , G is the gravitational constant, and m is the mass of the planet. Assume Gm = 1. When $x (0) = 1, x' (0) = y (0) = 0, y' (0) = 1$ the motion is a circular orbit of radius 1 and period $2 蟺$ .

(a) The setting $x_{1} = x, x_{2} = x', x_{3} = y, x_{4} = y'$ expresses the governing equations as a first-order system in normal form.

(b) Using localid="1664116258849" $h = \frac{2 蟺}{100} 鈮� 0.0628318$ ,compute one orbit of this moon (i.e., do N = 100 steps.). Do your approximations agree with the fact that the orbit is a circle of radius 1?

Show that the equation ${(\frac{dy}{dx})}^{2} + y^{2} + 4 = 0$ has no (real-valued) solution.

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