Question

Can you answer questions 2-6 please? Thanks

This homework will be due on Wednesday Feb 19th in discussion. Question 1. A cell of some bacteria divides into two cells every 40 minutes. The initial population is 200 bacteria. Express the population after t minutes as a function of t. Question 2. Find the orthogonal trajectories of the family of curves y2 = 6cx. Question 3 Find the general solution of y" – 6y' +9y = 0 Question 4 Find the general solution of y" + 2y' + 2y = 0 Question 5 Find the general solution of y(4)(x) + y" () – 6y = 0 Question 6 Find the general solution of y'"' – y'' – y' +y = 5, given y = 5 as a particular solution.

Answer 1

Solution Orthogonal trajectories of the family of curve 92=6cm Given that y ²6cn - 1) Differentiate equ (1) wisit n we get 29. mai = 66 (am) = 95.com) dom= c i = 9.1=a} Now putting value of c in equation (1) we get 92 - 6. (27 d ulin y?= 2y.dnim y² any.de y = 2n. din — (ii) Now writing dn for dy in equnciii) we get differential equr of the orthogonal trajectories as given below y = 2n. by separable Variable ydy n Now integrate both side we get Sydy = (-2ndn dn 2 Ž = - 2:11? + C d h - - 12+C y = -2 n²+2c= (y² + 2n²= 20) Answer

Solution Given that y"-69'+99=0 Given equation can be written an. (D2-6D + 9 )y=0 Auxiliary equation is m26m +9 = 0 {m=Die man = m?(3+3) m + 9 =0 =) m?3m-3m +9 50 = m(m-3)-3(m-3)=0 - (m-3)(m-3) =0 = m = 3,3 [ Lece if rooth are reported Hence Solution is y=(6,+(inseng Answer I then soll is given by (C,+ lanean solution 4 I 9"+2 y'+2y=0 Given equation can be written as (D?+2D+2)yco Auxilary equation is m2+2m +2=0 m =- b ± √b2-4ac - = -1+ i = -2 1/4-8 = -2 154 - =-2+2; Hence solution is given by exng Acousn + B sin ßn] for imaginary roote Here 2 =-1 & B=1 (y= [Acorn + B sinn] Answer occur in pair

solution 5 Given that 94(1) +9"(n)-6y=0 This can be written as (D4 + D²-6)y=0 Auxilary equation is given by mitm?_6=0 Here m-re is a factor of mi+m? 6=0 This (m-12) [m3 +52 m² +3m +352] =0 (m.52) [m? (mara) +3 (maral] to (m.52) [[m2+3)(m +12] 20 ie. (m-52) (mara) (m2+3)=0 il m=52,-52, toi Thun sol sly = clevert C2 Even Cycosntlysinn) Ansur Solutions Given that y'l_yl_y't y=5 n o This can be written as D²D +1)y= 5 Auxilary equin 3 m² m+1=0 h mlm-1)-1(M-1)=0 im? (m-1)=0 = (m+l)(m-1)(m-1)=0 This Complementry function am= 11 - ~= ((, +(angen +(geon Also given particular Integral as Thus complete sol" = C.F + P. I i.l. 19 = (Citlangen +(3en 5 Answer ( 0