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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: Q20E (page 64)

In Problems 9鈥�20, determine whether the equation is exact.
If it is, then solve it.
$(\frac{2}{\sqrt{1 - x^{2}}} + ycos (xy)) dx + (xcos (xy) - y^{- \frac{1}{3}}) dy = 0$

Short Answer

Expert verified

The solution is $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$ .

Step by step solution

01

Evaluate the equation is exact

Here $(\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) dx + (x \cos (xy) - y^{- \frac{1}{3}}) dy = 0$

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

$M (x, y) = (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) N (x, y) = (x \cos (xy) - y^{- \frac{1}{3}})$

$\frac{鈭� M}{鈭� y} = (\cos (xy) - x y \sin (xy)) = \frac{鈭� N}{鈭� x}$

This equation is exact.

02

Find the value of F(x, y)

Here

M (x, y) = (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) F (x, y) = 鈭� M (x, y) dx + g (y) = 鈭� (\frac{2}{\sqrt{1 - x^{2}}} + y \cos (xy)) dx + g (y) = 2 \sin^{- 1} x + \sin (xy) + g (y)

03

Determine the value of g(y)

$\frac{鈭� F}{鈭� y} (x, y) = N (x, y) x \cos (xy) + g' (y) = (x \cos (xy) - y^{- \frac{1}{3}}) g' (y) = - y^{- \frac{1}{3}} g (y) = - \frac{3}{2} y^{\frac{2}{3}} + C_{1}$

Now $F (x, y) = 2 si n^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} + C_{1}$

The solution of the differential equation is $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$ .

Hence the solution isrole="math" localid="1664178541241" $2 \sin^{- 1} x + \sin xy - \frac{3}{2} y^{\frac{2}{3}} = C$

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Most popular questions from this chapter

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

$\frac{dy}{dx} - \frac{y}{x} = x^{2} \sin 鈥� 2 x$

In problems $1 - 8$ identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $y = G (a x + b)$ . $\cos (x + y) dy = \sin (x + y) dx$

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

$鈥� \frac{dy}{dx} = \frac{x^{2} - y^{2}}{3 xy}$

Question: In Problems , solve the equation.

$鈥� (\sqrt{\frac{y}{x}} + \cos 鈥� x) dx + (\sqrt{\frac{x}{y}} + \sin 鈥� y) dy = 0$

Question: In Problems 1-30, solve the equation

$(2 x - 2 y - 8) dx + (x - 3 y - 6) dy = 0$

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