# Hw6key

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Homework 6 Due 7/19/21 MATH 305 Name k @,\}, 1. (5) A population follows the logistic equation 4& = P (120 3P), with P (0) = 10. (a) Draw the phase portrait. 1. Find the critical points. f ( (20 - 2F> =0 Pz0 or Pzt0 2. Place the arrows of the phase portrait on the P-axis. 3. Include the lines for the critical points. 4. Sketch a typical solution curve that goes through P (0) = 10. P40 < ho-30 <o L Al = Fr <0 T U »,fixlgg o<PcYo=Po0 ; } C o Flee Lot f ho-30 ¢ /C%'/Pe,'g:: Pr. :—:>d/d AL 20 j ; : | =g 60 -0 40 30 20 -I0 @ 0 20 30 40 50 6 P ' U P<0 ac-3p 56 (b) Find P (t) by using the generic solution P (t) = m{ff&;@fi = ff? <g) a<lro b=3 , = (0 o Fo p(fijvéfilfiiﬂfifi S22 TGy 4 (1re ()0 P(t): g%?alé;, 1200 - /aro = e mRe T e s 3o tgce 3 #9'6"{1'76 (c) | Carrying capacity = 40 (d) At what time will the population equal one-half the carrying capacity? Show your work. Write your answer in the box. \ _3¢] PLO= 4 (10) > 3550w 720 = Po=20G 7 /53'6@"?"{'"36'(}*"9 e = b Dot =l (k) €77 15 . 000
2. (3) Given the DE zy" + ¢/ + 25y =0 (a) Let IVP be the DE with the initial conditions y (2) = 4 and %/ (2) = 6. Find the largest interval containing zo = 2 such that Th. 4.1.1 guarantees that the IVP has a unique solution. Show your work. Write your answer in the box. Q "){*\'E = B % > NS ED? )L 504% condimvous on (-, o} &, (x )= 2 (-co 1) Y o comdinudind o, AR . Go ()= %=l or (10,c0) = IVP has G urigle 7 /'(k):' O C'Ow-g/mé{w&& s - g . v o e wotilion an (200) G (wy=%Z0 Lf;%av@/af thepm Yzp Interval = ( o { Cf) (b) Let BVP be the DE with the boundary conditions y (2) = 4 and y (4) = 8. 1. Does Th. 4.1.1 gurantee the existence of a solution on the interval you found in Part (a)? (circle one) Yes @ Th, 41 | Hdoes ot a/ap//kf to /f?é/f?g. 2. Does Th. 4.1.1 gurantee the uniqueness of a solution on the interval you found in Part (a)? (circle one) Yes @5 3. (1) Show your work. The functions =2z, fo(2) =2+ 3, f3 (x) = 6z + 6 are linearly (circle one): 1. Independent : Dependirvio/':} x4y (xtz)= Ex+é Ly talxez)-(exte]z0 R\Q(k}';iéfih(k)wr@ (j=o0 Led C,=2 C, 2 C31'~/ " 3 CLJ <, ¢ 3 G C\' (}') + o {;,,(;:;} Tj' G \(; ('%j =G & s 4. (1) Show your work. The functions f; (z) = 2z, fo (z) =z + 3, f3(z) = 2% + 2 are linearly (circle one): 1. Independé&d 2. Dependent Seppese Qd + e, £C) te, BE(x =0 g 2w ta Gz GO ) o CR 3 A (2¢ %C},},}fi'f + 35-)%'1(3 jakzq«éox.& o €3 O ; / <& % ! ' > A'; AT E Ty 2O —> G,z o . QJ.[XS & e //if'?é&(r{f) "'Géf?@?fiwdfff. ey 2, 420, 70 > =0