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Review Sheet for Exam II -- Answers Math 251, Spring 2007 2.2: 1c,d,e,j (these are in the textbook) Also: solve the initial value problem y'' – 2y' + 10y = 6 cos 3x – sin 3x; y(0) = 2, y'(0) = -8 2 ex cos 3x – (7/3) ex sin 3x – sin 3x 2.3: 1b,c,e; 2a,c (#1 in the textbook) 2a: yp = x sin x + cos x ln (cos x) 2c: yp = (1/2) cos x ln ( sec x + tan x) – (1/2) sin x ln (csc x + cot x) Also: solve the following two problems i) y'' – 2y' + y = (1/x) ex yh = A ex + B x ex + x ex ln (|x|) ii) y'' + y = sec x; y(0) = 1, y'(0) = 2 y = 2 sin x + 1 cos x + x sin x + cos x ln ( cos x) 2.4: 3, 6b, 8 (3 in textbook) 6b: yh = Ax + B/(x2) 8: y = Aex + B x2 ex 2.5: Be able to solve a problem like the following: Suppose a 4kg mass is attached to a spring with a spring constant of k = 20 N/m There is a frictional force in direction opposite to the motion given by Fm = b dx/dt. What value of b will result in critically damped motion? What type of motion will result if b is greater than this amount? less than this amount? Solns classified according to discriminant of r2 + (b/4) r + 20 Disc = (b2/16) – 80; b< 16 sqrt(5) underdamped, b = 16 sqrt (5) critically damped, b> 16 sqrt(5) overdamped 2.7: 3, 8, 16, 17 (3, 17 in textbook) 8: y = A sin x + B cos x + C sin 2x + D cos 2x 16: y = A + Bx + Cx2 + Dx3 + x4 + sin x p. 105: 2c,e; 4a,b 2c: y = -(9/8) e-x cos 2x + -(1/8) e-x sin 2x + (1/8) ex 2e: y = -(3/2) sin x + (5/2) cos x + (1/2) e-x 4a: y = A ex + B e-x + e2x 4b: y = A sin x + B cos x + sin 3x 3.2: 1, 2, 4, 6c (1 in textbook) 2: work with y’’ – (2/x) y’ + (2/x2) y = 0 for x not = 0 W(x) = x2 which is not =0 for x not= 0 Soln of initial value problem: y = x + 2 x2 4: W(x) = e4x, which is not=0 6c: W(x) = - e -5x, which is not=0 soln of initial value problem: y = 4 e-2x – 3 e -3x 3.4: Be able to do a problem like the following: Each of the following second order linear differential equations is in normal form. In each case show that solutions have infinitely many zeros on the positive x axis. i) y'' + (1/x) y = 0 ii) y'' + x2/(x3 + 1) y = 0 i) q(x) = 1/x. ii)q(x) = x2/(x3 + 1) In both cases, show the improper integral diverges. (Apply Theorem 3.6 on p. 136.) Consider the diff. eqn. y'' – 2y' + y = 0. Show the solutions y1 = aex and y2 = cex + dxex are linearly independent if and only if ad not = 0. W(x) = ad ex Non zero if and only if ad not = 0.