#Not OK to submit
Deven Singh, Assignment 3, 09/13/2021
#Q1
> L := [seq(Diff(y(t), t $ i) - y(t) = 0, i = 2 .. 10)];
> init1 := [y(0) = 1, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0, (D@@4)(y)(0) = 0, (D@@5)(y)(0) = 0, (D@@6)(y)(0) = 0, (D@@7)(y)(0) = 0, (D@@8)(y)(0) = 0, (D@@9)(y)(0) = 0, (D@@10)(y)(0) = 0];
> dsolve({L[1], init1[1], init1[2]}, y(t));
> sol2 := %;
> g2 := unapply(rhs(sol2), t);
> evalf(g2(1));
                          1.543080635
> dsolve({L[2], init1[1], init1[2], init1[3]}, y(t));
> sol3 := %;
> g3 := unapply(rhs(sol3), t);
> evalf(g3(1));
                          1.168058313
> dsolve({L[3], init1[1], init1[2], init1[3], init1[4]}, y(t));
> sol4 := %;
> g4 := unapply(rhs(sol4), t);
> evalf(g4(1));
                          1.041691470
> dsolve({L[4], init1[1], init1[2], init1[3], init1[4], init1[5]}, y(t));
> sol5 := %;
> g5 := unapply(rhs(sol5), t);
> evalf(g5(1));
                          1.008333609

> dsolve({L[5], init1[1], init1[2], init1[3], init1[4], init1[5], init1[6]}, y(t));
> sol6 := %;
> g6 := unapply(rhs(sol6), t);
> evalf(g6(1));
                          1.001388891

> dsolve({L[6], init1[1], init1[2], init1[3], init1[4], init1[5], init1[6], init1[7]}, y(t));
#Stops computing at k=6, unable to continue to k=10
#As k increases, a(0) -> 1
#Q2
> a := proc(n) option remember; if n = 0 then 2; else a(n - 1)^2; end if; end proc;
> a1 := proc(n) option remember; if n = 0 then 2; else 2^(2^n); end if; end proc;
> seq(a(i), i = 1 .. 25);
> seq(a1(i), i = 1 .. 25);
> evalb(% = `%%`);
true
> f := proc(n) option remember; if n = 0 then 6; else a(n - 1)^2; end if; end proc;
> a2 := proc(n) option remember; if n = 0 then 6; else 3*2^(2^n); end if; end proc;
> seq(f(i), i = 1 .. 25);
> seq(a2(i), i = 1 .. 25);
> evalb(% = `%%`);
true
#Q3
> rsolve({x(0) = 2, x(1) = 3, x(n) = 3*x(n - 1) - 2*x(n - 2)}, x(n));
1+2^n
#Q4
> rsolve({y(0) = 3, y(1) = 2, y(2) = 6, y(n) = 2*y(n - 1) + 2*y(n - 2) - 2*y(n - 3)}, y(n));
#outputs incomprehensible answer. unable to solve with Maple
#Q5
> rsolve({z(0) = 1, z(1) = 0, z(2) = 0, z(3) = 0, z(n) = z(n - 4)}, z(n));
((-1)^n)/4+1/4+((-i)^n)/4+((i)^n)/4