#ATTENDANCE QUIZ FOR LECTURE 3 of Dr. Z.'s Math336 Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p3
#with an attachment called
#p3FirstLast.txt
#(e.g. p3DoronZeilberger.txt)

#Right after attending the lecture, but no later than 4:00pm that day

Name: Julian Herman

Q1: Is the SIR model named after Sir Ronald Ross or does it stand for something else?
A1: No, it is named for its 3 possible states of members of a population with a contagious disease: Susceptible, Infected, Recovered (or alternatively, Removed).



Q2: Why is the lemma (if y1(t) is a solution, then c*y1(t) is also a solution for any constant c) NOT true for y'(t) = y(t)^2?
A2: Because in the above non-linear differential equation, c gets squared on the RHS but not on the LHS, thus the LHS is NOT equal to the RHS: 

d(c*y1)/dt = (c*y1)^2		#LHS: c comes out of derivative
c*dy1/dt = c^2*y1^2
dy1/dt = c*y1^2 IS NOT EQUAL TO y1^2... Differential equation is NON-LINEAR



Q3: a1:=the fifth digit of your RUID (204007768). If it's 0, make it 1; a2:=the first digit; a3=second digit. Solve by hand and maple: a1*y''(t)-a2*y'(t)+a3*y(t)=0, y(0)=0,y'(0)=0
A3: y''(t)-2*y'(t)+y(t)=0, y(0)=0, y'(0)=0
y(t)=e^(rt)
r^2*e^(rt)-2*r*e^(rt)+e^(rt)=0
e^(rt)*(r^2-2r+1)=0
(r^2-2r+1)=0
(r-1)^2=0
r1=1, r2=1
y1(t)=e^(t)	y2(t)=te^(t)

y(t)=c1*e^t + c2*t*e^t			y(0)=c1=0
y'(t)=c1*e^t + c2*e^t + c2*t*e^t		y'(0)=c2=0

y(t)=0... no non-zero solutions!



Q4: Why is the property that if a(n) is a solution, so is c*a(n) not valid for the non-linear recurrence: a(n)=a(n-1)^2
A4: Since the recurrence is non-linear, the solution space is specific and not general. 
For instance: 
Let a(n) be a solution, then:
c*a(n)=(c*a(n-1))^2
a(n)=c*a(n-1)^2 is not a solution because it IS NOT EQUAL to a(n-1)^2.