2.10. Šolve the following equations: (a) Yk+1 = (k² + 1)yk» (b) Yk+1 = (k – k² – k³)yk, (c)體=尝, Yk+1 k+1 Yk k
2.10. Šolve the following equations: (a) Yk+1 = (k² + 1)yk» (b) Yk+1 = (k – k² – k³)yk, (c)體=尝, Yk+1 k+1 Yk k
Related questions
Question
Solve the question in the same way as a book
Transcribed Image Text:2.10. Solve the following equations:
(k² + 1)yk,
= (k – k? – k*)yk,
(a) Yk+1
(b) Yk+1
-
(c) = .
Yk+1
k+1
Yk
k
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 = 1. The exact solution is given by the
expression
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 = 2yk
- 1
(2.179)
has a fixed point yk
1 and its exact solution is
Yk = 1+ A2*,
(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
= 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
