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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: Q22E (page 64)

In Problems 21鈥�26, solve the initial value problem.
$({ye}^{xy} - \frac{1}{y}) dx + ({xe}^{xy} + \frac{x}{y^{2}}) dy = 0, y (1) = 1$

Short Answer

Expert verified

$e^{xy} - \frac{x}{y} = e - 1$

Step by step solution

01

Evaluate the equation is exact

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

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

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

Therefore this equation is exact.

02

Find the value of F (x,y)

Here

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

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

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

.

Apply the initial conditions. $y (1) = 1$

$e^{1} - \frac{1}{1} = C C = e - 1$

Therefore, the solutions is role="math" localid="1664180725870" $e^{xy} - \frac{x}{y} = e - 1$ .

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Most popular questions from this chapter

Question: In Problems 31-40, solve the initial value problem.

$(t + x + 3) dt + dx = 0, 鈥夆赌夆赌夆赌� x (0) = 1$

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

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

Question: Riccati Equation. An equation of the form (18) $\frac{d y}{d x} = P (x) y^{2} + Q (x) y + R (x)$ is called a generalized Riccati equation.

If one solution鈥攕ay, u(x)鈥攐f (18) is known, show that the substitution $y = u + \frac{1}{v}$ reduces (18) to a linear equation in v.
Given that $u (x) = x$ is a solution to $\frac{d y}{d x} = x^{3} {(y - x)}^{2} + \frac{y}{x},$

use the result of part (a) to find all the other solutions to this equation. (The particular solution $u (x) = x$ can be found by inspection or by using a Taylor series method; see Section 8.1.)

In problem $1 - 6,$ determine whether the differential equation is separable $\frac{d y}{d x} = 4 y^{2} - 3 y + 1$ .

Question: In Problems , solve the equation. $鈥� (4 {xy}^{3} - 9 y^{2} + 4 {xy}^{2}) dx + (3 x^{2} y^{2} - 6 xy + 2 x^{2} y) dy = 0$

See all solutions

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