Page 349 - Advanced engineering mathematics

10.6 Phase Portraits 329

20 Ω 25 Ω
8. Find the currents in each loop of the circuit of
Figure 10.10. Assume that the currents and charges
are all zero prior to the switch being closed
at t = 0. 1/50 F
10 H 25 Ω
i 1 E out
40 Ω
i 2
45 V
i 3
10 –3 F FIGURE 10.12 Circuit for Problem 10, Section 10.5.

10 H
i 1 11. Derive a system of differential equations for the dis-
i 2
5 V placement functions for the masses in Figure 10.13, in
√
which a = 10 26. Assume that the top weight is low-
ered one unit and the lower one raised one unit, then
both are released from rest at time 0. The upper weight
FIGURE 10.10 Circuit for Problem 8, Section 10.5.
is free of external driving forces, while the lower
weight is subjected to an external force of magnitude
9. In the circuit of Figure 10.11, assume that the cur- F(t) = 39sin(t).
rents and charges are all zero prior to the switch
being closed at time 0. Find the loop currents for
time t > 0.
50 Ω
k = 65 – a
1
= 5
m 1
y 1
–3
10 F
i 1 10 H k = a
2
i 2
5 V
m = 13
2
y 2
k = 65 – a
3
FIGURE 10.11 Circuit for Problem 9, Section 10.5.
10. Find the loop currents in the circuit of Figure 10.12
for t >0, assuming that the currents and charge are all FIGURE 10.13 Mass/spring
zero prior to the switch being closed at t = 0. Also
determine the maximum value of E out (t) and when system for Problem 11,
this maximum value is reached. Section 10.5.

10.6 Phase Portraits

10.6.1 Classification by Eigenvalues
Consider the linear 2 × 2system X = AX with A as a real nonsingular matrix and

x(t)
X(t) = .
y(t)
We know how to solve this system. However, now we want to focus on the geometry and
qualitative behavior of solutions.

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October 14, 2010 20:32 THM/NEIL Page-329 27410_10_ch10_p295-342