Page 348 - Advanced engineering mathematics

328 CHAPTER 10 Systems of Linear Differential Equations

Brine Brine
5 gal/min 3 gal/min 10 gal/min

Tank 1 Tank 2

6 gal/min 2 gal/min 9 gal/min
FIGURE 10.6 Tank system for Problem 3, Section 10.5.

brine solutions are interchanged between the tanks
and also flow out of both tanks at the rates shown.
Determine the amount of salt in each tank for t ≥ 0.
Also calculate the time at which the brine solution k = 8
1
in tank 1 reaches its minimum salinity (concentration
of salt) and determine how much salt is in tank 1 at m 1 = 1/2
that time. y 1
4. Find the currents i 1 (t) and i 2 (t) in the circuit of
k = 3
2
Figure 10.7 for t > 0, assuming that the currents and
charges are all zero prior to the switch being closed at
t = 0. m = 1/2
y 2 2
50 Ω FIGURE 10.8 Mass/spring
system for Problems 5 and 6,
Section 10.5.

10 –3 F 7. Refer to the mechanical system of Figure 10.9. The
i 1 1 H left mass is pushed to the right one unit, and the right
mass is pushed to the left one unit. Both are released
5 V i 2 from rest at time t = 0. Assume that there are no
external driving forces. Derive and solve the differ-
ential equations with appropriate initial conditions for
the displacement of the masses, assuming that there
FIGURE 10.7 Circuit for Problem 4, Section 10.5.
is no damping. Denote left to right as the positive
direction.
Each of Problems 5 and 6 refer to the system of
Figure 10.8. Derive and solve the differential equations for
the motions of the masses under the assumption that there
is no damping.
k = 8 k = 5 k 3 = 8
2
1
5. Each mass is pulled downward one unit and released m = 2 m = 2
2
1
from rest with no external driving forces.
6. The masses have zero initial displacement and veloc-
ity. The lower mass is subjected to an external driving
FIGURE 10.9 Mass/spring system for Problem 7,
force of magnitude F(t) = 2sin(3t), while the upper
Section 10.5.
mass has no driving force applied to it.

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October 14, 2010 20:32 THM/NEIL Page-328 27410_10_ch10_p295-342