only 3 Differential Equations question needs to be done by MapleDepartment of Mathematics UAB

Differential Equations

MA252 Spring 2021

ASSIGNMENT 3

Due date: Tuesday 02/09/21

Linear First Order Equations

1. Find, by hand, showing all your working, all solutions of the first order linear equation

cos(x)

dy

+ sin(x) y = 2 cos3 (x) sin(x) − 1,

dx

0≤x<
π
.
2
Be careful here, as the equation is not in “standard form”.
2. A 60 kilogram skydiver (more correctly termed a “skyflopper”, as the picture below
indicates) jumps from a plane at a height of 1000 meters with an initial velocity of
v(0) = 0 meters per second, and falls subject to air resistance whose magnitude is
given by 15|v|, for 25 seconds. At this time his parachute opens and the air resistance
is now 180|v|. Assume that the positive direction is downward and the gravitational
acceleration constant is g = 9.8 m/s2 , and describe fully how you derived the relevant
differential equations. Use MAPLE to find his speed at impact, and the time at which
that occurred, and discuss whether or not he made it safely to the ground. Note that
it is known1 that survivability decreases √
markedly for fall heights h above 4 meters or
so, corresponding to a terminal velocity 2gh ≈ 9 meters per second.
All of the MAPLE commands needed for this assignment may be found in the MAPLE
file ass3-spring19.mw which is available from Canvas.
1
Kasim Turgut et. al., Falls from height: A retrospective analysis. World Journal of Emergency Medicine,
2018, 9(1), 46–50
3. An environmental hydrologist needs to know the total input, C, measured in cubic feet
per minute, of water into a shallow circular lake with known parabolic cross-section
y = 0.00001x2 , as indicated by the figure below, where the lake is formed by rotating
the parabola around the vertical axis. This quantity C is difficult to measure directly,
as the lake is fed by both surface run-off and underground streams, and depleted by
evaporation (also measured in cubic feet per minute) equal to 0.01 times the area of
the lake surface at any time. The hydrologist is seeking your mathematical help to
find C.
Figure 1: Circular lake vertical cross-section: y = 0.00001x2
10
9
8
7
6
y
5
4
3
2
1
0
−1,000
−500
0
500
1,000
x
(a) If the depth of water at the center of the lake is h feet find the volume V of water
in the lake, as a function of h.
(b) If we assume that the depth h is changing with time, find a differential equation for
the function h(t). Hint: find two different formulae for V 0 (t) the rate of change of
the lake volume, one from part (a) above, and the other by considering the input
and evaporation effects together.
(c) The hydrologist tells you that the depth h is approximately constant, at 10 feet,
over the late spring and early summer. Use this fact together with the differential
equation in (b) to inversely find C.
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