Homework for Lecture 2 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Sept. 13, 2021. Subject: hw2 with an attachment hw2FirstLast.pdf or hw2FirstLast.txt

1. A sequence is defined in terms of the recurrence, and initial conditions
	an = 4an−1 − 6an−2 + 4an−3− an−4 ,	a0 = 0	,	a1 = 1	.	a2 = 8,	.	a3 = 27,
(i) Find an for 1 ≤ n ≤ 8. (ii) Can you guess an explicit formula for an in terms of n? (iii) Can you prove it?
You are welcome to use Maple for the computations. You can write a short Maple code for a(n) using ”option remember”.
a := proc(n) option remember; 
if n = 0 
then 0; 
elif n = 1 
then 1; 
elif n = 2 
then 8; 
elif n = 3 
then 27; 
else 4*a(n - 1) - 18*a(n - 2) + 4*a(n - 3) - a(n - 4); 
end if; 
end proc;
seq(a(i), i = 1 .. 8);
1, 8, 27, -32, -583, -1656, 3715, 42368

2.	Solve the following differential equation, subject to the given initial condition.
Do it both by hand and using Maple (as we did in Lecture 2).
dsolve({y(0) = 1, D(y)(t) = (y(t)^3)/(t+1)}, y(t))
1/(sqrt(1-2ln(t+1)))

3.	Solve the following differential equation, subject to the given initial conditions
	y00(t) − 3y0(t) + 2y(t) = 0	,	y(0) = 2	,	y0(0) = 3	.
Do it both by hand and using Maple (as we did in Lecture 2)
dsolve({D(D(y))(t) - 3*D(y)(t) + 2*y(t), y(0) = 2, D(y)(0) = 3}, y(t))

4.	Find all the eigenvalues and corresponding eigenvectors of the matrix
Do it both by hand and using Maple (as we did in Lecture 2).
Λ = 7
V(1) = K(1)[1        -1]
Λ = -1
V(2) = K(2)[+1      -1]
Eigenvectors(Matrix([3,-4][4,3]));