#OK to post Homework #Jeton Hida, September 13, 2021, Assignment 3 >#Number 1 > dsolve({D(D(y))(t)-y(t)=0, y(0)=1,D(y)(0)=0},y(t)) 1 1 y(t) = - exp(-t) + - exp(t) 2 2 > evalf(exp(-1)/2 + exp(1)/2) 1.543080635 > dsolve({D(D(D(y)))(t)-y(t)=0, y(0)=1,D(y)(0)=0,D(D(y))(0)=0},y(t)) 1 2 / 1 \ /1 (1/2) \ y(t) = - exp(t) + - exp|- - t| cos|- 3 t| 3 3 \ 2 / \2 / > evalf(exp(1)/3 + (2*exp(-1/2)*cos(sqrt(3)*1/2))/3) 1.168058313 > dsolve({D(D(D(D(y))))(t)-y(t)=0, > y(0)=1,D(y)(0)=0,D(D(y))(0)=0,D(D(D(y)))(0)=0},y(t)) 1 1 1 y(t) = - exp(-t) + - exp(t) + - cos(t) 4 4 2 > evalf(exp(-1)/4 + exp(1)/4 + cos(1)/2) 1.041691470 > dsolve({D(D(D(D(D(y)))))(t)-y(t)=0, > y(0)=1,D(y)(0)=0,D(D(y))(0)=0,D(D(D(y)))(0)=0,D(D(D(D(y))))(0)=0},y(t)) / 256 exp(t) 1 |/ (1/2) \ y(t) = ------------------------------- + ----------------- |\16 5 + 16/ / (1/2)\ / (1/2)\ / (1/2)\ \ \40 + 8 5 / \40 - 8 5 / 5 \40 + 8 5 / / (1/2) \\ (1/2) // 1 (1/2) 1\ \ |1 (1/2) / (1/2)\ || 5 exp||- - 5 - -| t| cos|- 2 \5 - 5 / t|| - \\ 4 4/ / \4 // / / 1 | (1/2) / (1/2)\ //1 (1/2) 1\ \ |1 (1/2) ----------------- |5 \16 - 16 5 / exp||- 5 - -| t| cos|- 2 / (1/2)\ \ \\4 4/ / \4 5 \40 - 8 5 / (1/2) \\ / (1/2)\ || \5 + 5 / t|| // > evalf(256*exp(1)/((40 + 8*sqrt(5))*(40 - 8*sqrt(5))) + (16*sqrt(5) + > 16)*sqrt(5)*exp((-sqrt(5)/4 - 1/4)*1)*cos(sqrt(2)*sqrt(5 - > sqrt(5))*1/4)/(5*(40 + 8*sqrt(5))) - sqrt(5)*(16 - 16*sqrt(5))*exp((sqrt(5)/4 > - 1/4)*1)*cos(sqrt(2)*sqrt(5 + sqrt(5))*1/4)/(5*(40 - 8*sqrt(5)))) 1.008333609 > dsolve({D(D(D(D(D(D(y))))))(t)-y(t)=0, > y(0)=1,D(y)(0)=0,D(D(y))(0)=0,D(D(D(y)))(0)=0,D(D(D(D(y))))(0)=0,D(D(D(D(D(y)) > )))(0)=0},y(t)) 1 1 1 / 1 \ /1 (1/2) \ y(t) = - exp(-t) + - exp(t) + - exp|- - t| cos|- 3 t| 6 6 3 \ 2 / \2 / 1 /1 \ /1 (1/2) \ + - exp|- t| cos|- 3 t| 3 \2 / \2 / > evalf(exp(-1)/6 + exp(1)/6 + exp(-1/2)*cos(sqrt(3)*1/2)/3 + > exp(1/2)*cos(sqrt(3)*1/2)/3) 1.001388891 > dsolve({D(D(D(D(D(D(D(y)))))))(t)-y(t)=0, > y(0)=1,D(y)(0)=0,D(D(y))(0)=0,D(D(D(y)))(0)=0,D(D(D(D(y))))(0)=0,D(D(D(D(D(y)) > )))(0)=0,D(D(D(D(D(D(y))))))(0)=0},y(t)) #Maple unable to compute when k>=7. > #Only pattern I begin to notice is we are getting closer and closer to 1. > #Number 2 a:=proc(n) option remember: > if n=0 then 2: else a(n-1)^2: fi: end: > seq(a(n),n=0..5) 2, 4, 16, 256, 65536, 4294967296 > seq(a(n)-2^(2^n),n=0..10) 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 > #The constant multiple of the sequence a1(n) given by a2(n) = 3* 2^(2^n) is > not also a solution since in this case we have a nonlinear recurrence > equation, and our property with solution to linear recurrence equations where > if a1 is a solution then so is C*a1 does not apply here. > #Number 3 > rsolve({f(n)-3*f(n-1)+2*f(n-2)=0, f(0)=2, f(1)=3}, f(n)); n 1 + 2 > #Number 4 > rsolve({b(n)=2*b(n-1)+2*b(n-2)-2*b(n-3), b(0)=3, b(1)=2, b(2)=6},b(n)) Error, (in genfunc:-rgf_expand) unable to compute coeff > #Number 5 > rsolve({b(n)-b(n-4)=0,b(0)=1,b(1)=0,b(2)=0,b(3)=0},b(n)) 1 n 1 n 1 1 n - (-I) + - I + - + - (-1) 4 4 4 4 >