#Please do not post homework
#Julian Herman, 9/13/21, Assignment 3

#1
f := proc(k) option remember; diff(y(t), '`$`'(t, k)) - y(t) = 0; end proc;


dsolve({f(2), y(0) = 1, D(y)(0) = 0}, y(t));
                         1           1       
                  y(t) = - exp(-t) + - exp(t)
                         2           2       

dsolve({f(3), 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         /

dsolve({f(4), 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       

dsolve({f(5), 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         |/   
y(t) = ------------------------------- + ----------------- |\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) 
  5      + 16/ 5      exp||- - 5      - -| t| cos|- 2      
                         \\  4          4/  /    \4        

              (1/2)  \\                     /          
  /     (1/2)\       ||           1         | (1/2) /  
  \5 - 5     /      t|| - ----------------- |5      \16
                     //     /        (1/2)\ \          
                          5 \40 - 8 5     /            

                                         /         
         (1/2)\    //1  (1/2)   1\  \    |1  (1/2) 
   - 16 5     / exp||- 5      - -| t| cos|- 2      
                   \\4          4/  /    \4        

              (1/2)  \\
  /     (1/2)\       ||
  \5 + 5     /      t||
                     //


dsolve({f(6), 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         /


dsolve({f(7), 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));

#The 7th order won't evaluate, taking too long!

evalf(exp(-1)/2 + exp(1)/2);
                          1.543080635

evalf(exp(1)/3 + (2*exp(-1/2)*cos(sqrt(3)/2))/3);
                          1.168058313

evalf(exp(-1)/4 + exp(1)/4 + cos(1)/2);
                          1.041691470

evalf(256*exp(1)/((40 + 8*sqrt(5))*(40 - 8*sqrt(5))) + (16*sqrt(5) + 16)*sqrt(5)*exp(-sqrt(5)/4 - 1/4)*cos(sqrt(2)*sqrt(5 - sqrt(5))/4)/(5*(40 + 8*sqrt(5))) - sqrt(5)*(16 - 16*sqrt(5))*exp(sqrt(5)/4 - 1/4)*cos(sqrt(2)*sqrt(5 + sqrt(5))/4)/(5*(40 - 8*sqrt(5))));
                          1.008333609

evalf(exp(-1)/6 + exp(1)/6 + exp(1/2)*cos(sqrt(3)/2)/3 + exp(-1/2)*cos(sqrt(3)/2)/3);
                          1.001388891

#The trend is that the value of y(1) approaches 1!



#3
rsolve({a(0) = 2, a(1) = 3, a(n) = 3*a(n - 1) - 2*a(n - 2)}, a(n));
                              n    
                             2  + 1




#4
rsolve({a(0) = 3, a(1) = 2, a(2) = 6, a(n) = 2*a(n - 1) + 2*a(n - 2) - 2*a(n - 3)}, a(n));
               -----                 /                          n\
                \                    |  /     2           \ /1 \ |
                 )                   |  \-4 _R  - 4 _R + 3/ |--| |
                /                    |                      \_R/ |
               -----                 |- -------------------------|
           /    3       2           \|    /    2           \     |
_R = RootOf\2 _Z  - 2 _Z  - 2 _Z + 1/\    \6 _R  - 4 _R - 2/ _R  /




#5
rsolve({a(0) = 1, a(1) = 0, a(2) = 0, a(3) = 0, a(n) = a(n - 4)}, a(n));
                  1  n   1     n   1     n   1
                  - I  + - (-I)  + - (-1)  + -
                  4      4         4         4