#OK to post Homework
#Jeton Hida, September 13, 2021, Assignment 2
> #Number 1 (i) 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)-6*a(n-2)+4*a(n-3)-a(n-4): 
fi: 
end:
> seq(a(i),i=1..8);
                       1, 8, 27, 64, 125, 216, 343, 512
> #ii #Guess the explicit formula a(n)=n^3 

#iii Prove that a(n)=n^3, let's run program a(n)-n^3 if output is all 0's our guess is correct
> seq(a(n)-n^3,n=1..8);
                            0, 0, 0, 0, 0, 0, 0, 0
> #Number 2 dsolve({D(y)(t)=y(t)^3/(t+1), y(0)=1}, y(t))
                                         1           
                        y(t) = ----------------------
                                                (1/2)
                               (1 - 2 ln(t + 1))     
> #Number 3 dsolve({D(D(y))(t)-3*D(y)(t)+2*y(t)=0, y(0)=2,D(y)(0)=3},y(t))
                           y(t) = exp(2 t) + exp(t)
> #Number 4 A:=Matrix([[3, -4],[4, 3]])

> 
Loading LinearAlgebra

> with(LinearAlgebra):
> Eigenvectors(A)

> Vector[column]([3 + 4*I, 3 - 4*I]), Matrix([[I, -I], [1, 1]])

#Eigenvalues are in left vector, and corresponding eigenvalues are in the matrix next to it