AK-MATH225F10-WS2

AK-MATH225F10-WS2 - MATH225, Fall 2010 Name: \ Worksheet 2...

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Unformatted text preview: MATH225, Fall 2010 Name: \ Worksheet 2 (1.1, 1.2, 1.3) Section: S :2le 045 . - - ‘ i . . i ‘ ” A A i : F01 full credit, you must show all work and box answers. Lar76c‘f‘: ch 1— , - . ‘1“ a ,(y) :5? +q.[x)}jx 1. For the following, state the order of the differential equation and determine whether it is linear or nonlinear. [ $6404 * afflu- (a) (2 — fly” ~35fy' + 5t 2: sin(t) / (2-1:); I/ + i”3t> 7’ ‘— Simi’L)‘ S t Liam; aLLt):2~t/ outt) w-Bt/ mJt) =0, jac : smétrst 3,1 . Lister, sccywiws/‘alC/‘l (b) -— +y=0/ \ 4 ,,,, __ FTF‘th'oF/{J (“Tr/lax 5 “509 = 00/ («I-100:0 / «300 z X/ 014K): 0/ 61.00: l/GxoOOSO/ ?(x)=< (C) y :y / 7I+ (‘0):0 ‘ “w, «WWNu Linc: r~ For (F AL Tit” “i Q’ :1 ’ 6‘“ t ‘l/ :O / J'LKlan/ ‘Pmstnra’cr dy (l_ g \ (Cl) —=2y :)/}/=Z>~ %—®1wflllackr dt ( Mon hack/3 F7Pst"of‘Jc/‘ l 2. Verify that thv indicated function (or family of functions) is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. (a) y’ = 2y; W) = 62‘ ’(t); alt ‘6 1‘ L A! l -.‘ figs: 7 b t ' 7ft) =‘azt 75 at 50(dt’0’7 a FkS; l7lzdlqs (b) (y—w>y':y—:z:+& y<zz)=w+4¢ifii : Nani)? ..\7;’()<).=., ) +10%) 2 Ms: (7~x))’= WH ha? y() ( l + = Hr‘mz 4-3 “‘5 37-X+% ZMW —X+$/ :L. W+§g :(71X)7f/;QL\5 (c) g$g—4%+4y=0g 3/2016217—1—0256621’ 71x) 2y Him [\5 q soil /\ 2X 14 2X 7 ~ZC.c; Fab; +Zczxé 70: L’lC—lC/ZX’)‘ aczgx—iZCZ @ZX4. H CLXCZ 0‘5” 7hqy+q7 2%“+Q7K€K+WQ ‘1 . ‘ Maw {+92%} +Wfl ,1] ,, 9 :0 I r“1’6 ; Z2< Zxc (d)(.1:2—1U)""£+2:17y=0, —2;L"y+y‘=1/ IMP/roe S's/Q 7 C/C “FCLXCI lj 0t $o/Q (/J‘ $(“ZXL> +71) :j'lg'lg v2— 1 a“ \ —L+><7 ~széfi- +27%: :0 X 7‘2 r r an! h L ;\ (“if IC' 500 on "Zfiéx ~7) ’ 2(a)) C/ ka : szw7)A—;§ +—7,2<7:o 2 Ms X 3. Given y 2 C161” + 62x62”r is a family of solutions of y" — 4'9" + 4y 2 0. Find a solution of the second-order IVP (initial—value problem) consisting of this differential equation and the following initial conditions. (a) 31(0) = 0: 51/(0) = 0 2x zx , - 2x :2 61x 7 =c.¢ + sza / 7 -=. 4cm, +» Cap ZCZX 0 7' l I” :- ,'f 2: 7“ 9 0., 7 (O) .C (C) 31(1) = 7L 0. :1/(0)=~l 1) =¢,é+c2_e,z=o 7/[6>:ZC/+Ca:" ClCZ+LvI~ZCDCKIO CL: *I'ZC—l ‘ C45} : 62 (.25) / 7 :‘C +ch C]: "i 4. Determine if the following statement is true or false for each initial value. There is guaranteed to ex1st a unique solution to the initiaj value problem, where d—“lt/ = \/ 4 — y2 and the initial condition is given below. (a) y(0)=0 .___'}’_ro_c_ ¥£t,7) 1 (L! y ) :W C.<>n{7? nooOj fits? m3: 5. Given the differential equation y’ = y2, ar+c 7/:—()\+c)“‘ _ ;Z_~ 4—, 2 75 £H5:7/-+ZX+C) ' Zx+CJL 7 X‘ré- Q . - ' Z - l _ / ‘ gélu tho/1. Flafi. 2’ (kt) - éx+¢)z ‘ ,Qbs (b) Determin whether the differential equation is guaranteed to have a unique solution through the point (0.1). fCA/>): 73 Continuous Pol“ X «rial vodocg/ [Xe/Idem 3f = Z ConfTioOoOS (err X pmJ ValUcS/ 209 )lXe/Z,;H7Z3 2—7 ) > 9C5 W'kc pliicfo’cafii-xl G7UK€fc>o Uflfafitc \ ft) Agi/C a, Unbuci relation {3cm 0m initial Conalrfon/ IqCngl/j; [Ca/1:), (c) Find a solution of the first—o "der IVP (initial—value problem) consisting of this differential equation and the initial condition y(0) = 1. (a) Verify that y = is a one—parameter family of solutions. (d) Determin al I of definition for the solution of the inital-value problem in part (X) = :‘E‘ ACRE-ch 'mfio’l Jiffcftntffiblc For Xii j X! / ax/y>|/X4_IE Gabi The solution “is olefide on 1544 fr 7 I ; -r X ‘ infirm 01C :4ch ac», 6P tAaf vac/14M X / 6. A bacteria culture grows at a rate proportional to its size. {A3 Liniin Ccn‘bar‘fis (» fiT‘t! Ll ’\m I (a) Write thv differential equation that models this situat L011. Let P be the number of bacteria at time, t. Is this equation linear? (£9 >KP l / 235544sz was (b) Find an «'xplicit solution to your differential equation from part(a). (Hint: What function do you know rom calculus m such it is its own derivative? What__function has a derivative that is a nonstanLtJmes' ‘mitselfllémwwwm-M W 61 am 2 KOLE ms 47;: ilk/45' (“ks 1 [p #0,“?le :2“ Flag ‘. iLf’ fill/lo“; 321‘s (c) Now assume that the bacteria are exposed to a poison. The rate at which the bacteria is being killed by the poison is .51’2. Change the differential equation from part a to take into account this new factor, but do not solve this urw ODE. 7. (a) Suppose :| student carrying a flu Virus to an isolated college campus of 5000 students. Determine a differential equation for the number of people $(t) who have contracted the flu if the rate atuehich the disease spreads i5 proportional to the product of the number of students who have the flu and the number of students h hz't' tb ‘dt 't. H't:Th l fctd t: h 1 ‘ " — w 0 1w no eetn ‘8):be O 1 ( 1110494? )eagsaw sewvcogaxr/inovtigegnfiegpcgigs . I Wu: A X '.0U/‘1!9<f‘ op no: AKVC 3%— 0< ' ~ ‘ 0 o I ‘ l7~Sooo+I x “6:11:14! . Ax ;/4X (Scarf/-9 JV; ékaufa "165 Lean axpcécol (b) What would be the initial condition for thqssituation " " \ X[0)=ll 8. A cup of coffev is initially 165" F and is left in a room with an ambient temperature of 75" F. Suppose that at time t = 0 it is cooling at a rate of 200 per minute. Assume that Newton‘s Law of Cooling applies: The rate of cooling is proportional to the difference between the current temperature and the ambient temperature. . . . . . L rrm (a) Write an inital—value problem that models the temper iture of the coffee. Let T represent the temfieratug of the coffee at time, t, in minutes. ...
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This note was uploaded on 02/08/2011 for the course MATH 225 taught by Professor Staff during the Fall '08 term at Mines.

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AK-MATH225F10-WS2 - MATH225, Fall 2010 Name: \ Worksheet 2...

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