Theorem 3.1. The system (4) has a prime period-two solutions if and only if ? +€ -1 + 0, (22) 7 is true. Proof. First, assume that the system (4) has a distinct prime period-two solution in the follow- ing form ., (a1, B1), (a2, B2), (a1, 31), (a2, B2), ... where a + a2 and B1 + B2. Then, from (11) and (12), we obtain (@1 + a2) (B1 + B2) B2 – € (a1 + a2) (B1 + B2) B1 - € (@1 + a2) (B1 + B2) + Hy (23) a2 = + Hy (24) + €, (25) a2 – µ (a1 + a2) (B1 + B2) B2 = + €. (26) By some calculations from equations (23)-(26), we get 2 µ e (e – 1) e2 + € – 1 µ² (e? – (e + 1)) (e² +e – 1) (e² + € – 1)² - a1 + a2 = and a1 ¤2 = B1 + B2 = 1 and B1 B2 = 0, hence, we see that (a1 + a2)² – 4 a1 a2 # 0 and (ß1 + B2)² – 4 B1 32 # 0, which implies that a1 # a2 and ß1 # B2. Thus, µ (e? – (e + 1)) H (e² – e + 1) e2 + € – 1 , d2 = B1 = 0 and 6B2 = 1. (27) e2 + € – 1 Clearly a; exist if and only if e2 +e-1#0 and they are positive if e > 1 for i = 1, 2. Secondly, suppose that e² + € – 1 7 0 is satisfied. Then, the system (4) has a prime period two solution (e1, 5) = (" - (-+ 1),) and (a2, 52) = ("-+D). e2 + € – e2 + € – 1 Now, we show that (w1, z1) (a1, B1). From (4), we have (wo+ w_1) (z–1 + z-2) Wi %3D 20 – € (a1 + a2) (B1 + B2) + H, B2 - € (2εμ(ε-1) 1- € 1 + µ, e? + € – 1 2 e µ(e – 1) + µ (1 – e) (e² + e – 1) (1 – e) (e² + e – 1) 8
Theorem 3.1. The system (4) has a prime period-two solutions if and only if ? +€ -1 + 0, (22) 7 is true. Proof. First, assume that the system (4) has a distinct prime period-two solution in the follow- ing form ., (a1, B1), (a2, B2), (a1, 31), (a2, B2), ... where a + a2 and B1 + B2. Then, from (11) and (12), we obtain (@1 + a2) (B1 + B2) B2 – € (a1 + a2) (B1 + B2) B1 - € (@1 + a2) (B1 + B2) + Hy (23) a2 = + Hy (24) + €, (25) a2 – µ (a1 + a2) (B1 + B2) B2 = + €. (26) By some calculations from equations (23)-(26), we get 2 µ e (e – 1) e2 + € – 1 µ² (e? – (e + 1)) (e² +e – 1) (e² + € – 1)² - a1 + a2 = and a1 ¤2 = B1 + B2 = 1 and B1 B2 = 0, hence, we see that (a1 + a2)² – 4 a1 a2 # 0 and (ß1 + B2)² – 4 B1 32 # 0, which implies that a1 # a2 and ß1 # B2. Thus, µ (e? – (e + 1)) H (e² – e + 1) e2 + € – 1 , d2 = B1 = 0 and 6B2 = 1. (27) e2 + € – 1 Clearly a; exist if and only if e2 +e-1#0 and they are positive if e > 1 for i = 1, 2. Secondly, suppose that e² + € – 1 7 0 is satisfied. Then, the system (4) has a prime period two solution (e1, 5) = (" - (-+ 1),) and (a2, 52) = ("-+D). e2 + € – e2 + € – 1 Now, we show that (w1, z1) (a1, B1). From (4), we have (wo+ w_1) (z–1 + z-2) Wi %3D 20 – € (a1 + a2) (B1 + B2) + H, B2 - € (2εμ(ε-1) 1- € 1 + µ, e? + € – 1 2 e µ(e – 1) + µ (1 – e) (e² + e – 1) (1 – e) (e² + e – 1) 8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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