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Q16 E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 5

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 13-16, write a differential equation that fits the physical description. The rate of change of the mass A of salt at time t is proportional to the square of the mass of salt present at time t.

The differential equation suitable for the given condition is $\frac{dA}{dt} = A^{2}$ , where A is the mass of salt at a time t.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Analysing the given statement

By analyzing the given statement,

$\frac{dA}{dt} \propto A^{2}$ , where A is the mass of salt at a time t.

Step 2: Writing the differential equation

$\frac{dA}{dt} = {kA}^{2}$ , where k is the constant of proportionality.

Hence, $\frac{dA}{dt} = {kA}^{2}$ is the required differential equation.

Most popular questions for Math Textbooks

Find a general solution for the differential equation with x as the independent variable:

$y^{m} + 3 y^{n} + 28 y' + 26 y = 0$

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

$\frac{dy}{dx} = y^{4} - x^{4}, y (0) = 7$

(a) For the initial value problem (12) of Example 9. Show that $ϕ_{1} (x) = 0$ and $ϕ_{2} (x) = {(x - 2)}^{3}$ are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problem $y' = 3 y^{\frac{2}{3}},$ $y (0) = 10^{- 7}$ have a unique solution in a neighbourhood of $x = 0$ ?

Verify that the function $ϕ (x) = c_{1} e^{x} + c_{2} e^{- 2 x}$ is a solution to the linear equation $\frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} - 2 y = 0$ for any choice of the constants $c_{1}$ and $c_{2}$ . Determine $c_{1}$ and $c_{2}$ so that each of the following initial conditions is satisfied.

(a) $y (0) = 2, y' (0) = 1$

(b) $y (1) = 1, y' (1) = 0$

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

$y - \log_{e} y = x^{2} + 1$ , $\frac{dy}{dx} = \frac{2 xy}{y - 1}$

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