Math 336, Fall 2006
Test # 3
NAME .SOLUTIONS.
INSTRUCTIONS:
(1) Print your name and student number (ZID) above.
(2) Make certain that your test has all six (6) dierent sheets (including the cover page).
(3) You must SHOW YOUR WORK in order to get credit.
Math 336, Fall 2006
Quiz # 2
NAME .SOLUTIONS.
(1) Does Theorem 1 in Section 1.3 guarantee (local) existence and
uniqueness of solution of the IVP
dy
y
= x 1, y (0) = 1?
dx
Bonus question: What is/are the solution(s)?
f (x, y ) = x1 is continuous on its do
Math 336, Fall 2006
Quiz # 3
NAME .SOLUTIONS.
(1) Find the general solution (implicit if necessary, explicit if convenient).
y = xy 3 .
dy
= xdx;
y3
dy
=
y3
xdx + C ;
1
y 2 = x2 + C ;
y = (x2 C ) 2 .
Plus, the equilibrium solution: y = 0.
NOTE: Mind the i
Math 336, Fall 2006
Quiz # 4
NAME .SOLUTIONS.
(1) Solve the IVP with the logistic equation.
dx
= 4x(7 x), x(0) = 11.
dt
You may use the general formula for the solution as long as you
write it down explicitely and show me what you are substituting
for wha
Math 336, Fall 2006
Quiz # 5
NAME .SOLUTIONS.
(1) Suppose that a motorboat is moving 40 ft/s when its motor
suddenly quits, and that 10 s later the boat has slowed to 20
ft/s. Assume that the resistance it encounters while coasting is
proportional to its
Math 336, Fall 2006
Quiz # 6
NAME .SOLUTIONS.
(1) Find the general solution of
y + 8y + 25y = 0.
r2 + 8r + 25 = 0 r1,2 = 4 3i;
y (x) = e4x (c1 cos 3x + c2 sin 3x).
(2) Solve the IVP
3y (3) + 2y = 0,
y (0) = 1, y (0) = 0, y (0) = 1.
3r3 + 2r2 = 0, r = 0, 0
Math 336, Fall 2006
Quiz # 7
NAME .SOLUTIONS.
(1) Set up the appropriate form of a particular solution yp , but do
not determine the values of the coecients.
y (5) y (3) = ex + 2x2 5.
r5 r3 = 0, r = 0, 0, 0, 1, 1, yh = c1 + c2 x + c3 x2 + c4 ex + c5 ex .
Math 336, Fall 2006
Quiz # 8
NAME .SOLUTIONS.
(1) A mass m = 2 is attached to both a spring (with spring constant k = 50) and a dashpot (with damping constant c = 12).
The mass is set in motion with initial position x0 = 0 and initial
velocity v0 = 8. Fin
Math 336, Fall 2006
Quiz # 9
NAME .SOLUTIONS.
(1) Use partial fractions to nd the inverse Laplace transform of
5s 6
F (s) = 2
.
s 3s
5s 6
A
B
=+
;
2 3s
s
s
s3
5s 6 = A(s 3) + Bs = (A + B )s 3A; A = 2, B = 3.
3
2
+
= 2 + 3e3t .
f (t) = L1
s s3
(2) Use Lapl
Math 336, Fall 2006
Quiz # 10
NAME .SOLUTIONS.
(1) Find the convolution (f g )(t) if
f (t) = eat , g (t) = ebt , a = b.
t
t
a b(t )
(f g )(t) =
ee
0
(ab)
bt
d = e
e
bt
d = e
0
e (ab)
ab
t
=
=0
(2) Find Laplace transform of the function
f (t) = t if t 1;
Math 336, Fall 2006
Test # 1
NAME .SOLUTIONS.
INSTRUCTIONS:
(1) Print your name and student number (ZID) above.
(2) Make certain that your test has all ve (5) dierent sheets (including the cover page).
(3) You must SHOW YOUR WORK in order to get credit.
(
Math 336, Fall 2006
Test # 2
NAME .SOLUTIONS.
INSTRUCTIONS:
(1) Print your name and student number (ZID) above.
(2) Make certain that your test has all six (6) dierent sheets (including the cover page).
(3) You must SHOW YOUR WORK in order to get credit.
Math 336, Fall 2006
Quiz # 1
NAME .SOLUTIONS.
(1) Substitute y = erx into the equation y + y 2y = 0 and
determine all values of the constant r for which y = erx is a
solution of the equation.
y = erx implies y = rerx and y = r2 erx . Hence, the equation
b