practice exam

# practice exam - 1[ll Points Solve the equation dy 3 au=1 E...

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Unformatted text preview: 1. [ll] Points} Solve the equation dy— “3 au=1 E _ 1+ 172’ 2. {10 Points} Consider of 1 xz—y + rt: = —,y(1] =1 (II I 1’ al Determine the integrating faetor pk]. 2 pts (h) Rewrite the diﬁerential equation so that the left hand side is a perfect derivative. 2 pts {cl Solve for 3. {15 Points) Consider the following initial value problem % = 102? — SH .110) : C (a) (4 pts) Find the equilibrium solutions and determine whether they are stable nr nn‘l‘ 4 pts (b) {9 pts) Find the general and particular solution. Are there any singular solu— tions'? (e) [2 ptst Compute lin1¢_._Do xﬁ‘.) if 0 a: C' a: 2. 4. {1O Points) Draw the Slope ﬁelds and typicai solution curves for the equation 5. {10 Points) Find tho gonoral holntion of Tho {lifforontial equation: 3;” + 43f + 203,: = U. [5. {1L} Pointhj (a) ['5 13125] Find tho square root of —5 + 12 i. (b) ['5 13th] Solve :2 — E? r + E = {II and PKlH'IIhh the hﬂlllt ion no 3‘ + in. T. (15 Points) Leaves in a forest fall and accumulate on the ground at a rate of r kg per year. At the same time. the fallen leaves decompose at a rate proportional to the leaf mass on the ground. 1:vitli proportionality factor on Assume that the constants r, a .1» l]. (a) (2 pts) Let raft] be the leaf mass on the ground. IGive the differential equation for mltl. (b) (3 pts) lCJ'ver a long period of time. the leaf mass approaches an equilibrium value i‘l-f. Find M in terms of the annual accumulation rate r and the decomposition factor a. (c) (1D pts) A flood washes away all the leaves. How long does it take for the leaf mass to recover to 50% of its equilibrium value? 00 . (1D Points] A paratrooper bails out of an airplane at a height of 1.5 kin and opens her parachute at a height of [1.5 km above the ground. It is known that the limiting velocity of a free falling lmman body is 50 meters per second. Take air resistance to be proportional to the square of the velocity. [In this problem you may assume that gravitational constant is lﬂ mf-fs2.] {a} [2 pts] write down the differential equation governing the dynamics of the paratrooper. (b) (5 ptst Solve the differential equation. (c) [3ptsl How much time did she spend in the air before opening the parachute? ‘3. (10 Points) (a) (6 pts] Find the general solution of y“) + lesl + y" + 43,.” + 41;: = D. (b) (4 pts) Given that pm] = —1. y’fﬂ) = y"(f}] = yl3lfﬁl = 1. solve the initial value problem. 10. (10 Points) (3.) (6 pts) Using "Wronskian decide whether the following three functions are linearly.r independent fﬂx) = e‘. = 5—1. = (3—21. (h) (4 pts) Solve the following initial value problem 1:3 + 2y" — tr’ — 23,; = 0, 11(0) = 1. 31TH] = 2. ETD) = 0, given that f. g, and h. from part (a) are some solutions to the differential equation. ...
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