Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Question: Consider the initial value problem $\frac{dy}{dx} + \sqrt{1 + \sin^{2} xy} = x, y (0) = 2$ .
(a) Using definite integration, show that the integrating factor for the differential equation can be written as $μ (x) = (\int_{0}^{x} \sqrt{1 + \sin^{2} tdt})$ and that the solution to the initial value problem is $y (x) = \frac{1}{μ (x)} \int_{0}^{x} μ (s) sds + \frac{2}{μ (x)}$
(b) Obtain an approximation to the solution at x = 1 by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of $μ (x)$ and, thereby, the value of $\int_{0}^{1} μ (s) ds$ .
**[Hint: First, use Simpson’s rule to approximate $μ (x)$ at x = 0.1, 0.2, . . . , 1. Then use these values and apply Simpson’s rule again to approximate $\int_{0}^{1} μ (s) ds$ ]
(c)** Use Euler’s method (Section 1.4) to approximate the solution at x = 1, with step sizes h = 0.1 and 0.05. [A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]

0.9960
0.9486
0.9729

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1(a): Find the value of y(x)

Since $P (t) = \sqrt{1 + {sin}^{2} t}$ is a continuous real valued function on an open interval which contains points 0 we can use fundamental theorem of calculus to obtain

$\frac{d}{dx} [\int_{0}^{x} P (t) dt] = P (x) = \int_{0}^{x} P (t) dt = \int P (x) dx$

We can write integrating factor as $μ (x) = \exp (\int_{0}^{x} \sqrt{1 + \sin^{2} t} dt))$

Multiplying by $μ (x)$

$\frac{d}{dx} (μ (x) y (x)) = xμ (x) = dμ (x) y (x) = xμ (x) y (x)$

Now,

$μ (x) {y (x)}_{0}^{s} = \int_{0}^{s} (x μ (x)) = μ (s) y (s) - μ (0) y (0) = \int_{0}^{s} x μ (x) μ (s) y (s) = \int_{0}^{s} x μ (x) d x + 2 y (s) = \frac{1}{μ (s)} \int_{0}^{s} x μ (x) d x + \frac{2}{μ (s)}$

Step 2: Find the values of μ(x)

Here $y (s) = \frac{1}{μ (s)} \int_{0}^{s} x μ (x) d x + \frac{2}{μ (s)}$

Put x=1 then $y (1) = \frac{1}{μ (1)} \int_{0}^{s} x μ (s) d x + \frac{2}{μ (1)}$

We find the value of $\int_{0}^{1} μ (s) s d x$ .

Using Simpson’s rule

$\int_{a}^{b} f (x) d x = \frac{∆ x}{3} \sum_{k = 1}^{n} [f (x_{2 k - 2}) + 4 f (x_{2 k - 2}) + f (x_{2 k})] ∆ x = \frac{b - a}{2 n}, x_{k} = a + k ∆ x, k = 0.1, 0.2, . . ., 1$

So, $μ (x) = \exp (\int_{0}^{x} \sqrt{1 + \sin^{2} t} dt))$ at x=0.1,0.2, ……1

The values are

x = 0, =1

x = 0.1, =1.105354

x = 0.2, =1.223010

x = 0 .3, =1.355761

x = 0 .4, =1.506975

x = 0.5, =1.680635

x = 0 .6, =1.881401

x = 0 .7, =2.114679

x = 0 .8, =2.386713

x = 0 .9, =2.70670

x = 1, =3.076723

Now, using the previous conclusions

$∆ x = \frac{1}{2 n} = 0.1 = n = 5$

So,

role="math" localid="1663932388162" $\begin{array}{rcl} \int_{0}^{1} s μ (s) d s & = & \frac{0.1}{3} [0 \cdot μ (0) + 4 (0.1) \cdot μ (0.1) + 2 (0.2) \cdot μ (0.2) + 4 (0.3) \cdot μ (0.3) + 2 (0.4) \cdot μ (0.4) + 4 (0.5) \cdot μ (0.5) + 2 (0.6) \cdot μ (0.6) \\ + 4 (0.7) \cdot μ (0.7) + 2 (0.8) \cdot μ (0.8) + 4 (0.9) \cdot μ (0.9) + 1 \cdot μ (1)] \\ = & 1.064539 \end{array}$ role="math" localid="1663932537871" $\begin{array}{rcl} y (1) & = & \frac{1}{μ (1)} \int_{0}^{s} x μ (s) d x + \frac{2}{μ (1)} \\ = & 0.9960 \end{array}$

Therefore, $y (1) = 0.9960$

Step 3(b): Find the value of y(1) at h=0.1

The differential equation is $y' = x - \sqrt{1 + \sin^{2} x y}$

Use the recursive formula

localid="1663933001609" $\begin{array}{rcl} x_{n + 1} & = & x_{n} + h \\ y_{n + 1} & = & y_{n} + h f (x_{n} y_{n}) \\ = & y_{n} + h f (x_{n} - \sqrt{1 + \sin^{2} x_{n} y_{n}}) \end{array}$

Where,

$x_{0} = 0$ and $y_{0} = 0$ , h=0.1, N=10 steps at x=1

The values are

$x_{1} = 0.1, y_{1} = 1.8 x_{2} = 0.2, y_{2} = 1.6921 x_{3} = 0.3, y_{3} = 1.4830 x_{4} = 0.4, y_{4} = 1.3584 x_{5} = 0.5, y_{5} = 1.2526 x_{6} = 0.6, y_{6} = 1.1637 x_{7} = 0.7, y_{7} = 1.09 x_{8} = 0.8, y_{8} = 1.0304 x_{9} = 0.9, y_{9} = 0.9836 x_{10} = 1, y_{10} = 0.9586$

Therefore, $y = 0.9586$

Step 4 (c): Determine the value of y(1) at h=0.05

$x_{0} = 0$ and $y_{0} = 0$ , h=0.1, N=20 steps at x=1

$x_{1} = 0.05, y_{1} = 1.9 x_{2} = 0.1, y_{2} = 1.8074 x_{3} = 0.15, y_{3} = 1.4830 x_{4} = 0.2, y_{4} = 1.642 x_{5} = 0.25, y_{5} = 1.5683 x_{6} = 0.3, y_{6} = 1.5 x_{7} = 0.35, y_{7} = 1.4368 x_{8} = 0.4, y_{8} = 1.3784 x_{9} = 0.45, y_{9} = 1.3244 x_{10} = 0.5, y_{10} = 1.2747 x_{11} = 0.55, y_{11} = 1.229 x_{12} = 0.6, y_{12} = 1.187 x_{13} = 0.65, y_{13} = 1.149 x_{14} = 0.7, y_{14} = 1.1144 x_{15} = 0.75, y_{15} = 1.0831 x_{16} = 0.8, y_{16} = 1.0551 x_{17} = 0.85, y_{17} = 1.0301 x_{18} = 0.9, y_{18} = 1.0082 x_{19} = 0.95, y_{19} = 0.9892 x_{20} = 1, y_{20} = 0.9729$

Therefore, $y (1) = 0.9729$ $y (1) = 0.9729$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Step by Step Solution

Step 1(a): Find the value of y(x)

Step 2: Find the values of μ(x)

Step 3(b): Find the value of y(1) at h=0.1

Step 4 (c): Determine the value of y(1) at h=0.05

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Step by Step Solution

Step 1(a): Find the value of y(x)

Step 2: Find the values of μ(x)

Step 3(b): Find the value of y(1) at h=0.1

Step 4 (c): Determine the value of y(1) at h=0.05

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Calculus

Pure Maths