#Please do not post homework
#Julian Herman, 9/13/21, Assignment 2

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(i), i = 1 .. 8);
                1, 8, 27, 64, 125, 216, 343, 512

ii) the explicit formula is y(x)=x^3
iii) y(0)=0^3=0, y(1)=1^3=1, y(2)=2^3=8, y(3)=3^3=27 #Initial conditions verified.
n^3=?4*(n-1)^3-6*(n-2)^3+4*(n-3)^3-(n-4)^3 #Check with Maple

simplify(4*(n - 1)^3 - 6*(n - 2)^3 + 4*(n - 3)^3 - (n - 4)^3);
                                3
                               n 		#it equals n^3 ... proved.




2) dsolve({y(0) = 1, D(y)(t) = y(t)^3/(t + 1)}, y(t));
                                  1           
                 y(t) = ----------------------
                                         (1/2)
                        (1 - 2 ln(t + 1))     		#For by hand, see pdf




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)		#For by hand, see pdf




4)Eigenvectors(Matrix([[3, -4], [4, 3]]));
  [3 + 4 ⅈ]  [ⅈ  &uminus0;ⅈ]
  [                  ], [                                   ]
  [3 - 4 ⅈ]  [     1                  1          ]

Eigenvalues([[3, -4], [4, 3]]);
                      [3 + 4 ⅈ]
                      [                  ]
                      [3 - 4 ⅈ]			#For by hand, see pdf