# Review Sheet for Exam II -- Answers by carmeloanthony

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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.