Download Math 2930, Fall 2014, Prelim 1 Tuesday October 7, 7:30pm

Math 2930, Fall 2014, Prelim 1
Tuesday October 7, 7:30pm-9:00pm
Please note that no calculators, computers, etc. are allowed. One 1-sided, letter size
“formula sheet” may be used. Academic integrity on the part of the student is presumed violations will be dealt with swiftly and justly. Show your work: answer + reason = credit
0. (2 points) Write your name and section number on the front cover of your answer
booklet.
1. (12 points) True/False. Justify your answer.
(a) The equation y 00 + xy 0 = ey is a linear differential equation.
(b) The solution(s) for the initial value problem,
ty 00 + 3y = t, y(−1) = 1, y 0 (−1) = 2
is valid on (0, ∞).
(c) For any two differentiable functions f (x) and g(y), the separable equation
f (x) + g(y)y 0 = 0 is exact.
(d) If y1 and y2 are solutions of y 00 + y 0 + sin(y) = 0, then for any two constants
c1 and c2 , y = c1 y1 + c2 y2 is also a solution.
2. (16 points) Verify if y1 = x and y2 = xex are fundamental solutions of
x2 y 00 − x(x + 2)y 0 + (x + 2)y = 0
3. (20 points) Using any appropriate method solve the following equations (for general
solutions).
(a) xy 0 = 1 − (x + 1)y, x > 0.
y)
(b) y 0 = − y(x+ln
[Hint: Divide numerator and denominator of r.h.s by y].
x
(c) y 00 + 3y 0 = 0.
4. (25 points) Consider the differential equation,
y 0 + 2xy = −2xy 2
(1)
(a) Show that by using the substitution, u = y −1 the previous equation (1) can be
transformed to
u0 − 2xu = 2x
(2)
(b) Solve equation (2) for u.
(c) Using the result from (b), infer the solution y(x) of equation (1).
5. (25 points) Consider a conical water tank (with a cone angle φ) as shown in the
figure. Water is pumped in at a constant volume flow rate k and leaks out through a
small hole at the bottom of the tank. According to√
Torricelli’s law in hydrodynamics,
the rate at which water flows out is equal to α h, where α is a proportionality
constant and h is the current depth of water in the tank.
(a) Show that the depth of water in the tank at any time satisfies the equation
√
dh
(k − α h)
=
dt
πh2 tan2 φ
5. A cone
withthe
angletank
φ is partially
filled with
water as that
shown in
figure below.
A small find the equi(b) Assuming
that
is large
enough
it the
never
overflows,
drain hole is located at the bottom of the cone. Assume that the drain is initially shut
that at
t = 0 the height of the water in the cone is h . At t = 0 the drain
librium and
depth
oftime
water.
0
is opened and water begins to flow out of the cone. √
According to Torricelli’s law, the
volume of water remaining decreases at a rate of a h, where a is a proportionality
constant and h is the water height. Recall that the volume of a cone of height h and
radius r is V = πr2 h/3.
(c) Show that the equilibrium depth is asymptotically stable.
r(t)
φ
h(t)
fluid drain
Figure 1: Problem 5
(a) [15 points] Determine the differential equation and initial condition that when
solved will yield the height, h(t), of the water in the cone as a function of time.
(b) [5 points] Solve the differential equation.
(c) [5 points] In terms of φ, h0 and a, determine the time it will take to fully drain the
cone.
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