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DOC007 - Math 211 Exam 1 September 30th 2003 Instructions 0...

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Unformatted text preview: Math 211, Exam 1 September 30th, 2003 Instructions. 0 You have 75 minutes to complete the exam. Budget your time so that you will be able to attempt all sections. a Print your name and section on the EXAM BOOKLET. 0 Show all your work! Answers Without proper work will not receive full credit. 0 No calculators are allowed. 0 Put a box around your final answer. Upon finishing PLEASE write and Sign your pledge on front page of your exam booklet: On my honor I have neither given nor received any aid on this exam. ,1 t , "1 -~. JC, be.“ h (5w TV __ A, x... 3., a. l. (14 points) Find the exact solution of the initial value problem, and indicate the interval of existence. ( 2 ) , y +1 y = T y<1> = 2- 3;: [Q . (15 points) ' , . (a) Find the general solution to the differential equation Q ‘ '\ ty’ + y = 4232. (1)) Find the particular solution for the differential equation with initial value y(1) = 3. State the interval of existence of the solution. 3. (14 points) Consider the differential equation y’ = —2t‘/1 —- yz, Where \f‘ means the positive square root. (a) Is y(t) = sin(t2) a solution? ,“"‘\ (b) Is y(t) = (20509) a solution? WI, 1 . «4" 4. (14 points) Is it possible to find a function f (t, x) that is continuous and has contin— uous partial derivatives such that the functions 3:1(t) = t and 12(t) = sint are both solutions to r’ = f (t, at) near 75 = 0? Why or why not? Explain your reasoning. 5. (14 points) A tank contains 100 gal of pure water. A salt solution with concentration 3 lb/gal enters the tank at a rate of 2 gal/min. Solution drains from the tank at a rate of 2 gal /nn'n. Without solving the equation, use qualitative analysis to find the eventual concentration of the salt solutionhin the tank. '.: I M /,» .b 4" , 6. (14 points) For the initial value problem " y’ = ty, 21(0) =1, 7 " -;~ use Euler’s method to compute the first four iterations using step size h = 1/3. (i.e., calculate yo, y1, y2, and 93-) z. (15 points) Consider the autonomous equation ~ i ~ » y’=y(y-1)(y+2)v « ~:, ,1 (a) Find and classify all equilibrium points. (b) Draw the phase line. (c) Sketch equilibrium solutions on ty—plane. Sketch at least one solution trajectory in each of the regions on ty—plane divided by the equilibrium solutions. 5 WW, , . WW.-. - M...WWW--.»WWWWW-W_.WWWW-“WW...”WWW WWWW. ,_7_#MW,__WMW , ~ \ T ;§ / 0 Li" 56,4 x ‘ Lg ‘ ‘L , , [Am 9&3 _ 3‘ 5) Mile: small M I I LC \ ‘ 8 * I, 'J (\J b {2, "w-.. 7, W.-.” 5,-..1,“,mawahwmimw :;s t , LL, , . ..... h“, ,,.,,,__<~,---A--~ Q9?Efi.!r?u i” Lem: L 11 (432-5 “$9,“; ., _ / ,Mw, , 3‘ ¢ at i 6:1" \ ,5 LVQS - Tere (14 I“ ’ W ¢-- -,7#.»7A, ,r,_,,,H,, x x z ’ _. , .. ,. WHWWWme...“ MAHAWM,W..MW,HW...WWW W , WW, .. W ,.W 0 > ,x . W Nu V W q; x W W . u + _ W It 1 W / k , 4 “Mr WWW 1.1 L Q _ ii I ,I ‘ yr irWY, ‘ 141:!!! _ W é W W W . W W W W W, .W W W W W W, W W «W W W W W W“, W W. W, W W W .W W W W W.“ D “K ,4 W, W {A W_ W WWWWWiW W. W IV ‘ (X r W W + . W 4 W W pl! 4 1+ L. , W 0 ; W. W W W W e .23 W : w \ W9, W W, a W W 1v J I} W. W W : W r W L: . . , W W W t .W W t W W U}. W W W W m a. . W W W W MW) W W _W W W W W W W W W W W W W W Vbr I t . W 4+ II.» it! ,YIr WW+ \IWWYF >5 W W W W , W W W W W W W W W W ,W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W. W W W W W W W W W W W W W W W W W W W W W W W W W, , W W ,W W 4 W ...
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