Test1_11_Soln

Test1_11_Soln - 10’pt-s(1 Find all solutions to the...

Info icon This preview shows pages 1–11. 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
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 10’pt-s ' (1) Find all solutions to the following initial value problem. y should be mpressed as an explicit function oft. ty’ + 22527,: —- 47546":2 = 0, 11(0) 2 4 @011: y’ +21}; : 4439'“? We): 1th ; Qt‘v 'e‘ilyk My 949 QQtIy)’ ‘ : 4t? th/ °’ J4t34+ +C. 9% ”044w W :4 ' Q04 : OH, {OO 3 (2) Find all solutions to the following initial value problems. 3/ should be expressed as an explicit function oft. , 3t2+1_ __ g 27F+i \l ' 2%» 2w; +0 dy t 96+» )Ax J'égwata : \J <3€+wo€x+ L 8? >6 : “Cg-L325 45¢ Va) :4), 12 pts E E E E E E E E E E E E E E E E 10 pts (3) Find all solutions to the following initial value problem. y may be expressed as an implicit function of 9:. 2x + ycos(a:y) I -—-~———--—-—- = 0 1 = 1. + m.cos(a:y) + 33/2 ’ y( ) I : _ 2X+Y694pr \I XLoley)+%\r .lxwfiesmy) + [WEXYWWJ y’ .10 Lp ; x‘ + QTMXy) +£er ow ‘ , - W " XMW) l “05 “W ‘38; Um =3} if» : x‘a— Qa‘anys + %3 are/neml Lolfll‘é‘mi - xl-E anxy>+y3;{/ n'yu/‘el l+ EPA/EM :l‘, L: NEE/w v u <\\\-\\\1«W&4¥wmw\mmwzmww«rmwwfiw\ mm» a“ N swwxm mwm “ 5 (4) Find the general solution to the following differential equation. 3,] may be expressed as an implicit function of w. 2 xyy’—a:2 ~y2e (x) =0. XH= W1 95%;; y) : Xi+ +!i€vl) XY ,4L +5159 —L) y: She 12 pts 2 \DO (5) Multiplication of both sides of the following differential equa— tion by y“ for an appropriate value of a converts it to a type differential equation that we have studied: (3:264 +5y2’)y +4x3y=0==0. What is the appropriate value of a and what is the type of differential equation we have in mind? Prove your answer. 4 pts Do not solve the resulting differential equation. 4X$y yd + xgx4ya+xyl+d> \1/ :19 ‘ AW WW hm side/s (>5; 31 we Ga (6) The substitution 1) = y3 converts the differential equation 33:2y’ + my = 8”” into an equation equivalent with 7/ +p(g:)v = Q(x)v". ‘ - Find 10(90), q(m), and n. , , 6 pts y’ ; é” V~§ v’ xv’+ xv 23H? , i. ‘9} .3 v + x-V ~ ‘V - wad“ ne" “5%"? D. memmwmmwmmmn“ warmmxsmmwmwmymmwuwmmwxfiwsmmxavsmmwmcr:wawmmwwmmwmmwmxmmxfiwwfimfimfl 2 pts ‘4 pts (7) Consider the initial value problem (em -'1)2y’-ey3 = o, y<b> « 1. (a) Find a value I) such that the either the existence or the uniqueness (or both) of the solution is not guaranteed by the fundamental existence and uniqueness theorems. Justify yourX answer using these theorems funny 1 W 15w. {toque uolm @oM'M'dwlso ‘5 X «“1032 go new l? 0,11%le Nib/merm- M%11W5- (tr-term): CA wet QWQM ‘ (b) Attempt to solve the given initial value value problem for the value of b you found in part (a). Is there really a problem with either existence or uniqueness? Explain. Hint: f(e”c ~ l)”2e“c dx = ——(e”‘ —1)“1 + C , 21. 22 l) was 1, __L - QX J W0%7%\[email protected]»1f (fix "EC, 422.3"l ._ ~CQ££>4 «9L We)“ .L _ 1 I , “a ‘ (299—! M3 “ ‘ (8) Consider the initial value problem x+1’y(0):b' (a) Find, a value b such‘that the either the existence or the uniqueness (or both) of the solution is not guaranteed by the fundamental existence and uniqueness theorems. Justifyg our answer using these theorems. 2 pts -4 “E \ . film a \J’ AALL tk 4 .i ow“: y> X“ 3%; imxli > r T was? 3» =4, QM 95% Cl AXAWWM ad" v.14) go bit WWW bl‘ W myeaea; UM 52 0V (b) Attempt to solve the given initial value value problem for the value of b you found in part (a). Is there really a problem with either existence or uniqueness? Explain: 4 pts g _ with WW, Y (i _ 3’ 2 Q1739} : l/plX’l’ll +C/ .1 2, 2. U4 '4)? :fmfxmfi + E L, y l0) .: 4, C10 {VAC} bud‘%§/L-?/%Q 44$. \ZP’Dllé, . (9) Consider the following initial value problem. Without solv— ing it answer the following questions: 502—1 ’~ 21(1):?- _ 1+$2y2’ (a) Is there just one function y(a: )satisfying this initial value problem? Explain. _._y_...__, ”9% Sim-9- 399V “ll—7W) “Ml h§$w WEQ 40W (in, 34 My iv the aifiemw WWWMQ amen l0 ka Grivewe gelmmwww (b) On the basis of the fundamental existence and uniqueness theorem what can you say about the set of values 33 for which y)(x is defined? 4.. 5 m 9/ 91M, Wexfw sow) alwmelwmy .11 into/WE» l Wish/(fix WEE M9 max ATWl/d‘rjv \/L)<>\J mi WNW. (c) Assume that the solution exists for all :1}. For which values of m is y(m ) an increasing function? y HXR/ >0 is”. 0%) 3 PQM 5% WWW « 11 (10) A pond initially contains 106 gal of e h te bWater con— voy—0‘0! SM It 1% taining .03 + .01 sint lb/gal of pollut nts ows into the pond ‘ at 13,000 gal/day and flows out at the same rate. In addition, ‘3 9'5 9 a“! Mha‘ ‘ - ere is a house next to the pond that dumps 2 lbs of pollutants . J p the pond per day. Find an initial value problem (differ- 1‘” ”“6 ential equation and initial condition) that could be solved to find the pounds of salt Q(t) in the pond at time t. You may assume that the pollutants from the house are dumped at a constant rate and do not materially affect the volume of the pond. : Mo + gotta? +1 V» OLOB 52%} E f i E 10 pts W7) =36» létfiiw‘t - 030%») Oiqflzo/ i7 haw/pt?) ‘ [K We" X fifth”? ” ‘lbgl’lwwslébjj; ’ p~ 003+o\olémt)-Bwo +4 - l: H) X gees] 9%: l, l, i l .5 i 1 g g i i g 1' i l i l 8 pts 12 T Penna “t (11) Assume that an object of mass m kgs is thrown straight up from ground level with an initial velocity of 4m/sec. As- sume that air resistance produces a force of magnitude FW ~ .0001—|—+—L— Where v— —-.- u(t ) is the velocity at time t and :1:- — :ct( ) is the height above ground at time t. . (a) Write a second order differential equation. that could be‘solved to find a formula for x(t) during the period the object is rising. Your difierential equation should be totally in terms of x; no 12’s allowed. Do not solve the differential equation. 4" V3 \ 5 Mao; e l: 3W #0 1900‘ TH" \ V vs Pesgttve‘) (b) What are the appropriate initial conditions for the prob— lem 111 part? We) = 4 Owe/t [/f X 110 )Write a econd order differential equation that could )be solved to find a formula for x(t ) during the period the object is falling. Your dijj’erentz‘al equation should be totally in terms of 5c; no u ’3 allowed. Do not solve the differential equation. W\.Q\l F ‘1"— "OQEQ in“; if) 0“: “"3 ~ 0 00.91wa; ...... ...
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