1939.sagews

var('y')
MS = MatrixSpace(SR,2)
x_prime(x,y) =  x * ( 80 -x -y )
y_prime(x,y) = y * ( 120 -x-3*y -2*y^2)
f(x,y) = ( 80 -x -y )
g(x,y) = ( 120 -x-3*y -2*y^2)
fx(x,y) = derivative(x_prime(x,y),x)
fy(x,y) = derivative(x_prime(x,y),y)
gx(x,y) = derivative(y_prime(x,y),x)
gy(x,y) = derivative(y_prime(x,y),y)

def comm(x,y):
    A = x * fx(x,y) + f(x,y)
    B = x * fy(x,y)
    C = y * gx(x,y)
    D = y * gy(x,y) + g(x,y)
    return MS.matrix([A,B,C,D])

solve([x_prime(x,y)==0,y_prime==0],x,y)

[[x == 0, y == 0], [x == 80, y == 0], [x == 0, y == -1/4*sqrt(969) - 3/4], [x == 0, y == 1/4*sqrt(969) - 3/4], [x == 76, y == 4], [x == 85, y == -5]]

latex(solve([x_prime(x,y)==0,y_prime==0],x,y))

\left[\left[x = 0, y = 0\right], \left[x = 80, y = 0\right], \left[x = 0, y = -\frac{1}{4} \, \sqrt{969} - \frac{3}{4}\right], \left[x = 0, y = \frac{1}{4} \, \sqrt{969} - \frac{3}{4}\right], \left[x = 76, y = 4\right], \left[x = 85, y = \left(-5\right)\right]\right]

comm(0,0)

[ 80 0] [ 0 120]

comm(80,0)

[-6400 -6400] [ 0 40]

comm(0,1/4*sqrt(969) - 3/4)

[ -1/4*sqrt(969) + 323/4 0] [ -1/16*(sqrt(969) - 3)^2 -1/8*(sqrt(969) - 3)^2 - 1/32*(sqrt(969) - 3)*((sqrt(969) - 3)^2 + 2*(sqrt(969) - 3)*sqrt(969) + 6*sqrt(969) - 978) - 3/4*sqrt(969) + 489/4]

comm(0,-1/4*sqrt(969) - 3/4)

[ 1/4*sqrt(969) + 323/4 0] [ -1/16*(sqrt(969) + 3)^2 -1/8*(sqrt(969) + 3)^2 + 1/32*(sqrt(969) + 3)*((sqrt(969) + 3)^2 + 2*(sqrt(969) + 3)*sqrt(969) - 6*sqrt(969) - 978) + 3/4*sqrt(969) + 489/4]

comm(76,4)

[-5776 -5776] [ -16 -304]

comm(85,-5)

[-7225 -7225] [ -25 425]