math252a - Math 252 Exam 1 Musial Musial Name—_—...

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Unformatted text preview: Math 252 Exam 1 February 28, 2008 Musial Musial Name—______— Instructions. Work each problem on the exam.. Where you are asked to solve an equation, make sure to find all solutions. Show all work. 1. Solve the DE. j—Z=1+$+y+$y. % : My +7(/47(] M My jJ/M "3 "w '3 *C 2 ) - s'X'JrlgL'FC 79+ X2 kl Jr a - Q < )< 6. gal: 2. Solve the DE and give the largest intervals on which a solution can exist: 937/ + 2y = 3x .’ L -5 , 51%;?”3 7640 withg _ Q 7> 7” >21 3. A tank initially contains 300 gallons of saline solution with salinity 6 lbs of salt per gallon of solution. Solution having salinity 4 lbs of salt per gallon is pumped into the tank at a rate of 2 gallons per minute, and solution leaves the tank at the same rate of 2 gallons per minute. Find the function A (t) which models the amount of salt in the tank at any time, and find the amount if salt in the tank after a long time. 21 gay/M) L; lei/gig I/H 4. The number of students at a college which have been exposed to an influenza virus is modelled by the DE d2: E _. km (2000 — x) where a: is the number of students exposed after t days. On day 0, 1 student has been exposed, on day 6, 15 students have been exposed. Solve for k.. W $0000» "X2 ZKVH’ W \ , /—+——~— : _‘—« .1 . 1 7C ( 1000 ' X) 209° 74' A + goo / e “ :wwfix +A/waa-Y)/ 1 fowJA/Y/W.—>c)) So A %{0(‘/ma~>) .: 14 +c 3.00) 0+ éw/ ‘ 0+ += 6, ngo/Qq(("fiqq] : C J— /Qh(/S- lag?) 200 0 049038 f, GK +5003g K: O. 000 125/ 5. Verify that the following differential form is exact and solve it. Note that — arctan y = dy $+y (m+arctany)d$+1+y2dy=0. M W /V‘ N' \ 9N ‘ ”1 / "’" 1"" /I) A x)— - ' ‘ ’ 7 7L lev‘CJ-cmfl '1“ 42 Jh(/’Uj2) 1' C 6. Solve the DE 1 7. Verify that yl (x) = E is a solution to the following DE. Then use the reduction of order formula to find a second linearly independent solution. $231” + 3xy' + y = 0. j/Vlfl} :y—r j,({)c] : ,— XP' 2' 6f”): '1 f3 x Lj,” +— siw,’ + 3 -: i _ 3 ”X ”Sc “LB/C: 6”» / -jNoch ‘32 3 \JJ ix v3); 7C Y”?- A g -*\w~“ X x >4 X 8. Solve the DE 9. Solve the DE y" +4y'+ 3y = 0. ASSOSIM {lawn/w ml 4‘ Lin/1+ 3 :O (MALE/X/Mw/ia M: ~ 51m: v , my 6 3 ‘2 Q» < X\(LJ7L) 10. Solve the DE 3/” +4y’ + 53/ = 0. 10 ...
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