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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 1: Q9RP (page 1)

Decide whether the statement made is True or False. The relation $x^{2} + y^{3} - e^{y} = 1$ is an implicit solution to $\frac{dy}{dx} = \frac{e^{y} - 2 x}{3 y^{2}}$ .

Short Answer

Expert verified

The statement is false.

Step by step solution

01

Using the differential formula

For the result use the differential formula $\frac{d}{dx} (x^{n}) = n x^{n - 1}$ and consider x and y as variable.

02

Differentiating x2+y3-ey=1 with respect to x.

The given statement is,

$x^{2} + y^{3} - e^{y} = 1 \frac{d}{dx} (x^{2} + y^{3} - e^{y}) = \frac{d}{dx} (1) 2 x + 3 y^{2} \frac{dy}{dx} - e^{y} \frac{dy}{dx} = 0 (3 y^{2} - e^{y}) \frac{dy}{dx} = - 2 x \frac{dy}{dx} = \frac{2 x}{e^{y} - 3 y^{2}}$

Since, this is the given differential equation, the given statement is false.

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Most popular questions from this chapter

In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.

$\frac{dx}{dt} = 1, \frac{dy}{dt} = 3 x^{2}; x (t) = t + 1, y (t) = t^{3} + 3 t^{2} + 3 t$

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

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

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$e^{xy} + y = x - 1$ , $\frac{dy}{dx} = \frac{e^{- xy} - y}{e^{- xy} + x}$

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{dy}{dt} - ty = \sin^{2} t, y (蟺) = 5$

In Problems 10鈥�13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

$(1 + t^{2}) y'' + y' - y = 0; y (0) = 1, y' (0) = - 1 on [0, 1]$

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