only 3 Differential Equations question needs to be done by MapleDepartment of Mathematics UAB
MA252 Spring 2021
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
+ sin(x) y = 2 cos3 (x) sin(x) − 1,
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. Purchase answer to see full attachment
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