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Math 830-831, Qualifying Exam, January, 2005 1. Show that each of the following systems has a nontrivial periodic solution: (a) x0 = x + y − x(x2 + 2y 2 ), y 0 = −x + y − y(x2 + 2y 2 ). (b) x0 = x − y − x3 , y0 = x + y − y3. 2. Prove that µ0 is a Floquet multiplier of the Floquet system x0 = A(t)x iff there is a nontrivial solution x satisfying x(t + ω) = µ0 x(t) for all t. 3. Using the variation of constants formula solve the IVP 0 0 0 t x0 = 0 2 1 x + 0 , 0 0 2 0 1 x(0) = 1 . 1 4. Find an appropriate Green’s function and use it to solve the BVP (e−5t x0 )0 + 6e−5t x = e3t , x(0) = 0, x(log(2)) = 0. 5. Prove that the calculus of variations problem Z 1 Q[x] = {e−4t [x0 (t)]2 − 4e−4t x2 (t)} dt 0 subject to x(0) = 1, x(1) = 0 has a proper global minimum and find this minimum value. 6. Assume p, q, and r are continuous on [a, b), p(t) > 0 on [a, b) with limt→b− p(t) = 0. Prove that eigenfunctions corresponding to distinct eigenvalues of the SLP (p(t)x0 )0 + q(t)x = λr(t)x x, x0 are bounded on [a, b), γx(b) + δx0 (b) = 0, satisfy a certain orthogonality condition. γ 2 + δ2 > 0