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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: Q11E (page 64)

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

Short Answer

Expert verified

The solution is $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$ .

Step by step solution

01

Evaluate whether the equation is exact

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

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

$M (x, y) = (e^{x} \sin y - 3 x^{2}) N (x, y) = (e^{x} \cos y + \frac{y^{- \frac{2}{3}}}{3}) \frac{鈭� M}{鈭� y} = e^{x} \cos y = \frac{鈭� N}{鈭� x}$

This equation is exact.

02

Find the value of F (x, y)

Here

$M (x, y) = (e^{x} \sin y - 3 x^{2}) F (x, y) = 鈭� M (x, y) dx + g (y) = 鈭� (e^{x} \sin y - 3 x^{2}) dx + g (y) = e^{x} \sin y - x^{3} + g (y)$

03

Determine the value of g(y)

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

Now $F (x, y) = e^{x} \sin y - x^{3} + \sqrt[3]{y} + C_{1}$

The general solution of the differential equation is $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$

Hence, the solution is role="math" localid="1664171463132" $e^{x} \sin y - x^{3} + \sqrt[3]{y} = C$

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Most popular questions from this chapter

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

$2 {xy}^{3} dx - (1 - x^{2}) dy = 0$

Question: Use the substitution $v = x - y + 2$ to solve equation (8).

$\frac{dy}{dx} = y - x - 1 + {(x - y + 2)}^{- 1}$

In problem 7-16, solve the equation. $\frac{dy}{dx} = \frac{x}{y^{2} \sqrt{1 + x}}$

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

$鈥� (3 x^{2} - y^{2}) 鈥� dx + (xy - x^{3} y^{- 1}) dy = 0$

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

$鈥� [1 + {(1 + x^{2} + 2 xy + y^{2})}^{- 1}] dx + [y^{- \frac{1}{2}} {(1 + x^{2} + 2 xy + y^{2})}^{- 1}] dy = 0$

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