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Q27 E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 14

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.
$y \frac{dy}{dx} = x, y (1) = 0$

The hypotheses of Theorem 1 are not satisfied.

The initial value problem does not have a unique solution.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding the partial derivative of the given relation concerning y

Here, $f (x, y) = \frac{x}{y}$ and $\frac{\partial f}{\partial y} = - \frac{x}{y^{2}}$

Step 2: Determining whether Theorem 1 implies the existence of a unique solution or not

From Step 1, we find that $\frac{\partial f}{\partial y}$ is not continuous or even defined when $y = 0$ . Consequently, there is no rectangle containing the point $(1, 0)$ , in which both $f (x, y)$ and $\frac{\partial f}{\partial y}$ are continuous. Because the hypotheses of Theorem 1 do not hold, we cannot use Theorem 1 to determine whether the given initial value problem does or does not have a unique solution. It turns out that this initial value problem has more than one solution.

Hence, Theorem 1 implies that the given initial value problem does not have a unique solution.

Most popular questions for Math Textbooks

In Problems 10–13, 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.

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

Let $ϕ (x)$ denote the solution to the initial value problem

$\frac{dy}{dx} = x - y, y (0) = 1$

⦁ Show that $ϕ (x) = 1 - ϕ' (x) = 1 - x + ϕ (x)$

⦁ Argue that the graph of $ϕ$ is decreasing for x near zero and that as x increases from zero, $ϕ (x)$ decreases until it crosses the line y = x, where its derivative is zero.

⦁ Let x* be the abscissa of the point where the solution curve $y = ϕ (x)$ crosses the line $y = x$ .Consider the sign of $ϕ (x *)$ and argue that $ϕ$ has a relative minimum at x*.

⦁ What can you say about the graph of $y = ϕ (x)$ for x > x*?

⦁ Verify that y = x – 1 is a solution to $\frac{dy}{dx} = x - y$ and explain why the graph of $y = ϕ (x)$ always stays above the line $y = x - 1$ .

⦁ Sketch the direction field for $\frac{dy}{dx} = x - y$ by using the method of isoclines or a computer software package.

⦁ Sketch the solution $y = ϕ (x)$ using the direction field in part (f).

Spring Pendulum. Let a mass be attached to one end of a spring with spring constant k and the other end attached to the ceiling. Let $l_{o}$ be the natural length of the spring, and let l(t) be its length at time t. If $θ (t)$ is the angle between the pendulum and the vertical, then the motion of the spring pendulum is governed by the system

$l'' (t) - l (t) θ' (t) - gcosθ (t) + \frac{k}{m} (l - l_{o}) = 0 l^{2} (t) θ'' (t) + 2 l (t) l' (t) θ' (t) + gl (t) sinθ (t) = 0$

Assume g = 1, k = m = 1, and $l_{o}$ = 4. When the system is at rest, $l = l_{o} + \frac{mg}{k} = 5$ .

a. Describe the motion of the pendulum when $l (0) = 5 . 5, l' (0) = 0, θ (0) = 0, θ' (0) = 0$ .

b. When the pendulum is both stretched and given an angular displacement, the motion of the pendulum is more complicated. Using the Runge–Kutta algorithm for systems with h = 0.1 to approximate the solution, sketch the graphs of the length l and the angular displacement u on the interval [0,10] if $l (0) = 5 . 5, l' (0) = 0, θ (0) = 0 . 5, θ' (0) = 0$ .

Newton’s law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, $\frac{dT}{dt} = K [M (t) - T (t)]$ where K is a constant. Let $K = 0.04 {(\min)}^{- 1}$ and the temperature of the medium be constant, $M (t) = 293 kelvins$ . If the body is initially at 360 kelvins, use Euler’s method with h = 3.0 min to approximate the temperature of the body after

(a) 30 minutes.

(b) 60 minutes.

In Problems 13-19, find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

$(x^{2} + 1) y'' - e^{x} y' + y = 0; y (0) = 1, y' (0) = 1$

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