Exam1Solutions2006 - M427K Exam 1 Question 1[10 Points A...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M427K Exam 1 , February 21, 2006 Question 1 [10 Points] A 1000 L tank initially holds 200 L of solution of the extremely toxic Uranium Nitrate salt U 02(N O3)2 with an initial concentration of 2 ,ug/ L. Solution containing 8 pg/ L of U 02(N 03)2 salt enters the tank at the rate of 4L/min. Well mixed solution is allowed to flowout of the tank at the rate of 2 L/min. 1. [3 Points] Write an initial value problem describing the amount (in ug) of U 02(N 03)2 salt in solution after t minutes. ‘ , C\ d? _ A GLO gig-:- Rmxcm‘ Rom“ ea 3%-C5X4 1 1006“: (Mel =4OO 2. [5 Points] Solve the initial value problem to find the amount of U 02(N O3)2 salt in solution after t minutes . Sill.- gilfil as 151, [00% i at ‘ l d_Q EL: * 'DZOOi-t Gd =ln too+t O‘NW 31 PL 1 [y SO REC): IOO’rt ji- Qoma Q] = mow) [Omag = $100+, + [6&2 +C ra= W 9(0):; =3 C=4oooo C9 lOCH: \OO _ ltt1+3zoot #0000 QC» Moo 3. [2 Points] Find the moment that the tank is full. Find the theoretical limit— ing concentration that would result if the tank had infinite volume. Will the concentration at the moment that the tank is full be lower or higher than the theoretical limiting concentration ? You do not need to compute the concentration at the moment the tank is full. The tank I‘S [Ni oi t=400 WW8. l? H412 [QM hod m finite Volume and w [d at time 90 lo +oo Na would have 0 lwmimg - Conmnir‘olion oi Climilmg = Cm 1 8 93/1.- ‘ HM ; "M concaniroxlion 1‘) ~[own W4 #14 limihn Stimulmiglnu Ne NIL hot/2 [M conwnirqiion (firth? WOW/Li [We WW [S [W bang William the Womb ch[ 2 [MW- ContamimilOl/L M427K Exam 1 . ' February 21, 2006 Question 2 [10 Points] 1. [3 Points] Apply the Existence and Uniqueness theorem for first order linear ordinary differential equations to ty’ + y = tsin(t), WT) = 310 giving the largest interval for which solutiOns are guaranteed to exist and be unique. ’ i. Pitt (19‘) (1r . , h ' in standard Perm 8 + z 5 #smt. yti) and q ) are (on mom on Hue whole. real line excluding JC:O ler‘e pill hob 0 Wm mgmpluh. in large} MiHVCti animal/2:43 it loud QXCibLOimg V‘Q is COME) i Q)? giomimd firm i 2. [4 Points] Find the general solution of the ordinary differential equation . ty'+y=tsin(t) , . Art) = e‘"’“=L 0% [t8] =t5mi ' £3 =ft3midi ; «Most + it“ OR ., ~tcori + sm‘r +0 ._ i+ m + 0' @ W ‘ e t 3. [3 Points] Find the value of yo such that the solutiOn of the initial value problem ty’ + y = 15811105), . y(7r) = yo exists for all times t. Explain What happens for other values of yo. 3: —-C0_Si;+ int {9 OLQRYIQOi For 0“ t This gives BCWF—‘cosurh 3%“ :l :3 90:}, \9 96M ihcn (>6 and We solqhon 8012) )0 +00 068w (upfrooch ’90. 8mm the right > g ' l? aka-l "Mm 'CLO and HQ >olul10n3t>m is ~00 q) 301:? avr'wd‘ W “0"“ M “3” M427K . Exam 1 _ February 21, 2006 Question 3 [10 points] 1. [8 Points] Find an explicit solution to the ordinary differential equation @_4t3+1 dt 231—2- Sayorqble 533-014 = {ff-H, +C Q54? : t4+tiC.‘ ~l ziw ‘j — | iJJL4+JL+C 2. [2 Points] Find an explicit solution to the initial value problem dy_4t3+1 dt _ 2y—2’ if 1— th+t+4 21(0) = —1- M427K Exam 1 February 21, 2006 Question 4 [10 points] 1. [5 Points] Show that the first order ordinary differential equation (4:103 + 33/) doc + (4y3 + 3:13) dy = O is exact and hence find the general solution] You are nOt required to find the solution explicitly. mil‘f‘th SOexocl. , Cling? [email protected]+?>3) dx = X4 +318 + (13) Cbbfiij) 3 %8+3X)d3 —: a!“ lgxfl + DCX) erfimaweofiwr\'¢®8>=X““asrg* Or) and in general solubon u _ xq+3x3 +39 = C. 2. [5 Points] Find an integrating factor that makes the first order ordinary differ— ential equation (a: — yz)‘dm + 2mg dy = 0 into an exact equation. You are not required to find the solution. MS-NL=-4g M: -_’_l_ » defend) onlj on x180. can find 1 N x depending 0an on x C.“ M427K Exam 1 ' . February 21, 2006 Question 5 [10 points] y Solve the homogeneous first order ordinary differential equation dy _ x3+y3 E- 3my2 ' Give the solution explicitly if you can. 3 fl : High) 0W 305/1? Lei V= 3/x 36 HP X3311 +V = 151% fl_ W” _ 7C oix _ ”5‘41 V = W” 4%? . 3V1 3v7’ = 1““ V?’ V2 3‘3"" = -d>< Hv?’ X Z “214$ .3V 3 dv do _. vex/10W i'lV “lidu -.—. 5V1dV g "id” : ~--‘- In lot} = “dilly“ i'lvgi M fl ax: lnlxi+C --‘— .n\l~lv°’i= M M +0 7. . ' mung] = \n Ail )“1V8 2 1' ADC—1 1 I ‘1 V3 = I 1 A1 3 z ‘92:)CgiA/x M427K Exam 1 February 21, 2006 Question 6 [10 points] Consider the autonomous ordinary differential equation E = NJ) where f(y) = y4(y2 f 4)(y2 + 1)(y — 1)3- 1. [4 Points] Sketch f (y) vs y. Roois al 5=O 3:1?» Onol 3" W 2. [4 Points] Find the equilibrium solutions and classify each according to its stability. . Equilibrium solubom ore _ 3: 5L unsiablt ‘5 3 O semisiable ' ' 5 = l osgmpluhcollfl Slflbl€_ 3 a l uHsloble 3. [2 Points] Sketch y(t) vs t. ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern