#OK to post homework
#Deven Singh, 09/10/2021, Assignment 2
#Problem 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); end if; end proc;

> seq(a(n), n = 1 .. 8);
1, 8, 27, 64, 125, 216, 343, 512
# (ii) a(n) = n^3
# (iii)

> seq(a(n) - n^3, n = 1 .. 8);
0,0,0,0,0,0,0,0

> expand(n^3 - 4*(n - 1)^3 + 6*(n - 2)^3 - 4*(n - 3)^3 + (n - 4)^3);
0
#Problem 2

> dsolve({diff(y(t), t) = y(t)^3/(t + 1), y(0) = 1}, y(t));
y(t) = 1/sqrt(1-2*ln(t+1))
#Problem 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) = e^t+e^(2t)
#Problem 4

> A := Matrix([[3, -4], [4, 3]]):

> with(LinearAlgebra):

> evalf(Eigenvalues(A));
[[3+4i],[3-4i]]

> evalf(Eigenvectors(A))[2];
[i,1] and [-i,1]