MATH 251Final examMay 4, 2000Name:Section:There are 10 partial credit questions, each worth 15 points. In order to obtain fullcredit for these problems, all work must be shown. Credit will not be given foran answer not supported by work. The point value for each question is in parenthesesto the right of the question number.You may not use a calculator on this exam.1:2:3:4:5:6:7:8:9:10:Total:Do not writein the boxto the leftMATH 251 -Final exam-1. Solve the initial value problemy0= 2t + 2ty, y(0) = 3.Page 2 of 12MATH 251 -Final exam-2. Find the general solution to each of the following:(a) 4y00− 4y0+ y = 0(b) y00− 2y0+ 2y = 0(c) y00+ 5y0+ 6y = 0Page 3 of 12MATH 251 -Final exam-3. Find the form of a particular solution of the equation:y00+ 4y = t2et− 2 cos 2t + (t + 1) sin t.DO NOT solve for the constants.Page 4 of 12MATH 251 -Final exam-4. (a) Find the Laplace transform of f (t) =(0 : 0 ≤ t < 3t2: t ≥ 3.(b) Find the inverse Laplace transform ofs − 2s2+ 2s + 10.Page 5 of 12MATH 251 -Final exam-5. Solve the following initial value problem.y00− 2y0= 4δ(t − 1), y(0) = 0, y0(0) = 0Page 6 of 12MATH 251 -Final exam-6. Let A =−1 23 4.(a) Find the general solution of x0= Ax.(b) Classify the type and stability of the critical point at00(c) If x(0) =2αand limt→∞x(t) =00, then what is the value of α?Page 7 of 12MATH 251 -Final exam-7. Consider the following nonlinear system:x0= x2+ y2− 10y0= 2x − 6y(a) Find all the critical (fixed) points.(b) For each critical point, find the eigenvalues of the linearized system near that point.(c) What conclusions can you draw about the type and stability of the critical pointsof the nonlinear system?Page 8 of 12MATH 251 -Final exam-8. Separate the following partial differential equation into two ordinary differential equa-tions.2uxx+ uxt= 0Page 9 of 12MATH 251 -Final exam-9. Solve the heat conduction equation,4uxx= utu(0, t) = 0 = u(3, t) for t > 0u(x, 0) = sin(2πx3) − 2 sin(πx) + 7 sin(5πx3)Page 10 of 12MATH 251 -Final exam-10. Letf(x) =(0 : 0 ≤ x < 11 : 1 ≤ x < 2.(a) Sketch the even, period 4 extension of f(x) on the interval −6 ≤ x ≤ 6.(b) Find the Fourier series for the above extension.(c) What does the above series converge to when x = 0, x = −3/2, x = 1, and x = 2?Page 11 of 12MATH 251 -Final exam-f(t) = L−1{F (s)} F (s) = L{f (t)}11s, s > 0eat1s − a, s > atn, n a positive integern!sn+1, s > 0tp, p > −1Γ(p + 1)sp+1, s > 0sin atas2+ a2, s > 0cos atss2+ a2, s > 0eatsin btb(s − a)2+ b2, s > aeatcos bts − a(s − a)2+ b2, s > atneat, n a positive integern!(s − a)n+1, s > auc(t)e−css, s > 0uc(t)f(t − c) e−csF (s)ectf(t) F (s − c)f(ct), c > 01cF (sc)δ(t − c) e−csf(n)(t) snF (s) − sn−1f(0) − · · · − f(n−1)(0)Page 12 of
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