Q14E ... [FREE SOLUTION]

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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 3: Q14E (page 113)

Short Answer

Expert verified

Step by step solution

Given Data

We have given differential equation,

Using Kirchhoff law

Applying Kirchhoff law in loop (1) and (2)
we get,

Substitute values

Now in complete network

Substitute equation 3 in equation 1

Final answer

Now, Substitute equation (3) in (2)

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Most popular questions from this chapter

When forgetfulness is taken into account, the rate of memorization of a subject is given by

$\frac{d A}{d t} = k_{1} (M - A) - k_{2} A$

Where, $k_{1} > 0, k_{2} > 0, A (t)$ is the amount memorized in time t, M is the total amount to be memorized, and $M - A$ is the amount remaining to be memorized.

(a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of $A (t)$ as $t \to \infty$ . Interpret the result.

(b) Solve the DE subject to $A (0) = 0$ . Sketch the graph of $A (t)$ and verify your prediction in part (a).

Solve Problem 23 under the assumption that the solution is pumped out at a faster rate of $10 gal / \min$ . When is the tank empty?

Suppose that a glass tank has the shape of a cone with circular cross section as shown in Figure 3.2.10. As in part (a) of Problem 33, assume that $h (0) = 2 ft$ corresponds to water filled to the top of the tank, a hole in the bottom is circular with radius $\frac{1}{32}$ in., $g = 32 ft / s^{2}$ , and $c = 0.6$ . Use the differential equation in Problem 12 to find the height h(t) of the water.

(b) Can this water clock measure 12 time intervals of length equal to 1 hour? Explain using sound mathematics.

A heart pacemaker, shown in Figure 3.1.15, consists of a switch, a battery, a capacitor, and the heart as a resistor. When the switch Sis at P, the capacitor charges; when Sis atQ, the capacitor discharges, sending an electrical stimulus to the heart. In Problem 58 in Exercises 2.3 we saw that during this time the electrical stimulus is being applied to the heart, the voltageE across the heart satisfies the linearDE.

$\frac{d E}{d t} = - \frac{1}{R C} E$

(a) Let us assume that over the time interval of length the switch S is at position P shown in Figure 3.1.15 and the capacitor is being charged. When the switch is moved to position Q at time $t_{1}$ the capacitor dis-charges, sending an impulse to the heart over the time interval of length $t_{2}, t_{1} ⩽ t < t_{1} + t_{2}$ . Thus over the initial charging/discharging interval $0 ⩽ t < t_{1} + t_{2}$ the voltage to the heart is actually modelled by the piecewise-linear differential equation

$\frac{d E}{d t} = {\begin{cases} 0, 0 ⩽ t < t_{1} \\ - \frac{1}{RC} E, t_{1} ⩽ t < t_{1} + t_{2} \end{cases}$

By moving S between P and Q, the charging and discharging over time intervals of lengths $t_{1}$ and $t_{2}$ is

Repeated indenitely. Suppose $t_{1} = 4 s, t_{2} = 2 s, E_{0} = 12 V$ , and $E (0) = 0, E (4) = 12, E (6) = 0, E (10) = 12, E (12) = 0$ and so on. Solve for $E (t)$ for $0 ⩽ t ⩽ 24$ .

(b) Suppose for the sake of illustration that $R = C = 1$ . Use a graphing utility to graph the solution for the IVP in part (a) for $0 ⩽ t ⩽ 24 .$

Suppose an RC-series circuit has a variable resistor. If the resistance at time $t$ is defined by $R (t) = k_{1} + k_{2} t$ , where $k_{1}$ and $k_{2}$ are known positive constants, then the differential equation in (9) of Section 3.1 becomes