#OK to post homework

#Nicholas DiMarzio, 11/8/21, Assignment 18

#
#Problem 1
C:= proc(a,b,c,d,e) local x1,x2,x3,x4 option remember;
x1 := b/a
#x1 is number of eggs one chicken lays in c days;
x2 := x1/c
#x2 is the number of eggs one chicken lays in one day;
x3 := e*x2
#x3 is the number of eggs one chicken lays in e days;
x4 := d*x3
#x4 is the number of eggs d chicken lay in e days;
end:

C(3/2, 3/2, 3/2, 3, 3);
                               6
#
#Problem 2
#
W := proc(a, b, k) local g, vc2; option remember; 
g := solve(1/b + (k - 1)*vc = 1/a, vc);
print(1/g); 
end proc;

W(4, 5, 2);
                               20

#
#Problem 3
#
F := Matrix([[-2, -2], [-4, -2]]);
                      [     -2        -2]
                 F := [                 ]
                      [     -4        -2]

evalf(Eigenvalues(F));
                     [    0.828427124     ]
                     [                    ]
                     [.   -4.828427124]
#So (-2,1) is unstable

G := Matrix([[1, 0], [3, 0]]);
                               [1  0]
                          G := [    ]
                               [3  0]

evalf(Eigenvalues(G));
                              [0.]
                              [  ]
                              [1.]


G := Matrix([[6, 0], [8, 0]]);
                               [6  0]
                          G := [    ]
                               [8  0]

evalf(Eigenvalues(G));
                              [0.]
                              [  ]
                              [6.]

G := Matrix([[-4, 0], [-2, 0]]);
                           [-4  0]
                      G := [     ]
                           [-2  0]

evalf(Eigenvalues(G));
                         [    0.     ]
                         [           ]
                         [   -4.     ]

#Since y is any real number, we have to check and see if there is a trend to the stability.
#We can see here that no matter what y is, 0 is a eigenvalue of the matrix
#Therefore no matter what y is the equilbrium points are always unstable.

#Problem 4
#
Dis2([-x^2 - x*y + x, -2*x^2 - x*y + 3*x], x, y, [2.1, -0.9], 0.01, 10);

#We see that (2,-1) is unstable 

Dis2([-x^2 - x*y + x, -2*x^2 - x*y + 3*x], x, y, [0.1, 0.1], 0.01, 10);

#We see that (0,0) is unstable

Dis2([-x^2 - x*y + x, -2*x^2 - x*y + 3*x], x, y, [0.1, -4.9], 0.01, 10);

#We see that (0,-5) is unstable

Dis2([-x^2 - x*y + x, -2*x^2 - x*y + 3*x], x, y, [0.1, 5.1], 0.01, 10);

#We see that (0,5) is unstable
#We then can see that (0,y) where y is any real number, is unstable.

#Problem 5
Dis2(SIRS(s, i, 0.01, gamma, 1, 50), x, y, [50 - 30, 30], 0.01, 10);
Dis2(SIRS(s, i, 0.01, gamma, 1, 80), x, y, [80 - 30, 30], 0.01, 10);
Dis2(SIRS(s, i, 0.01, gamma, 1, 120), x, y, [120 - 30, 30], 0.01, 10);