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2.8.2 Example B Consider the following linear difference equation: 3yk+1 = Yk + 2. (2.177) 1) The exact solution is given by the This equation has a fixed point yk = expression

2.8.2 Example B Consider the following linear difference equation: 3yk+1 = Yk + 2. (2.177) 1) The exact solution is given by the This equation has a fixed point yk = expression

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2.8.2 Example B Consider the following linear difference equation: Зук+1 Yk +2. (2.177) This equation has a fixed point yk expression = 1) The exact solution is given by the Yk = 1+ A3¬k, (2.178) where A is an arbitrary constant. Consideration of both Figure 2.6 and the result of equation (2.178) shows that the fixed point is stable. The linear difference equation Yk+1 = 2ук — 1 (2.179) has a fixed point yk 1 and its exact solution is Yk = 1+ A2k, (2.180) where A is an arbitrary constant. For this case, the fixed point is unstable. See Figure 2.7. Likewise, the equation Yk+1 -2yk + 3 (2.181) has the fixed point Yk = 1. Since the slope is larger in magnitude than one, the fixed point is unstable. The exact solution is Yk = 1+ A(-2)*, (2.182) where A is an arbitrary constant. Note that for these three examples, we have, respectively, monotonic con- vergence, monotonic divergence, and oscillatory divergence. See Figures 2.6, 2.7, and 2.8.
Transcribed Image Text:2.8.2 Example B Consider the following linear difference equation: Зук+1 Yk +2. (2.177) This equation has a fixed point yk expression = 1) The exact solution is given by the Yk = 1+ A3¬k, (2.178) where A is an arbitrary constant. Consideration of both Figure 2.6 and the result of equation (2.178) shows that the fixed point is stable. The linear difference equation Yk+1 = 2ук — 1 (2.179) has a fixed point yk 1 and its exact solution is Yk = 1+ A2k, (2.180) where A is an arbitrary constant. For this case, the fixed point is unstable. See Figure 2.7. Likewise, the equation Yk+1 -2yk + 3 (2.181) has the fixed point Yk = 1. Since the slope is larger in magnitude than one, the fixed point is unstable. The exact solution is Yk = 1+ A(-2)*, (2.182) where A is an arbitrary constant. Note that for these three examples, we have, respectively, monotonic con- vergence, monotonic divergence, and oscillatory divergence. See Figures 2.6, 2.7, and 2.8.
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