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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 2: Q16E (page 64)

In Problems 9鈥�20, determine whether the equation is exact.
If it is, then solve it.
$({ye}^{xy} - \frac{1}{y}) dx + ({xe}^{xy} + \frac{x}{y^{2}}) dy = 0$

Short Answer

Expert verified

The solution is $e^{xy} - \frac{x}{y} = C$ .

Step by step solution

01

Evaluate whether the equation is exact

Here $({ye}^{xy} - \frac{1}{y}) dx + ({xe}^{xy} + \frac{x}{y^{2}}) dy = 0$

The condition for exact is $\frac{鈭� M}{鈭� y} = \frac{鈭� N}{鈭� x}$ .

$M (x, y) = ({ye}^{xy} - \frac{1}{y}) N (x, y) = ({xe}^{xy} + \frac{x}{y^{2}})$

$\frac{鈭� M}{鈭� y} = e^{xy} + x y e^{xy} + \frac{1}{y^{2}} = \frac{鈭� N}{鈭� x}$

This equation is exact.

02

Find the value of F (x, y)

Here

$M (x, y) = ({ye}^{xy} - \frac{1}{y}) F (x, y) = 鈭� M (x, y) dx + g (y) = 鈭� ({ye}^{xy} - \frac{1}{y}) dx + g (y) = e^{xy} - \frac{x}{y} + g (y)$

03

Determine the value of g(y)

$\frac{鈭� F}{鈭� y} (x, y) = N (x, y) ({xe}^{xy} + \frac{x}{y^{2}}) + g' (y) = ({xe}^{xy} + \frac{x}{y^{2}}) g' (y) = 0 g (y) = C_{1}$

Now $F (x, y) = e^{xy} - \frac{x}{y} + C_{1}$

The solution of the differential equation is $e^{xy} - \frac{x}{y} = C$ .

Therefore, the solution is role="math" localid="1664174122218" $e^{xy} - \frac{x}{y} = C$

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Most popular questions from this chapter

Question: In Problems 1-30, solve the equation.

$(y - x) dx + (x + y) dy = 0$

In problems 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $y^{'} = G (ax + by)$ .

$(y^{3} - {胃测}^{2}) 诲胃 + 2 胃^{2} ydy = 0$

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9-16.

$鈥� \frac{dx}{dt} = \frac{x^{2} + t \sqrt{t^{2} + x^{2}}}{tx}$

In problems identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form .

.

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 $渭 (x) = ({鈭�}_{0}^{x} \sqrt{1 + \sin^{2} tdt})$ and that the solution to the initial value problem is $y (x) = \frac{1}{渭 (x)} {鈭�}_{0}^{x} 渭 (s) sds + \frac{2}{渭 (x)}$

(b)Obtain an approximation to the solution at x= 1 by using numerical integration (such as Simpson鈥檚 rule, Appendix C) in a nested loop to estimate values of $渭 (x)$ and, thereby, the value of ${鈭�}_{0}^{1} 渭 (s) ds$ .

[Hint:First, use Simpson鈥檚 rule to approximate $渭 (x)$ at x= 0.1, 0.2, . . . , 1. Then use these values and apply Simpson鈥檚 rule again to approximate ${鈭�}_{0}^{1} 渭 (s) ds$ ]

(c)Use Euler鈥檚 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.]

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