Find all critical points of the given plane autonomous syste

Find all critical points of the given plane autonomous system. (Enter answers as a comma-separated list.)

Solution

Critical points are given by setting:x\'=y\'=0

x\'=0 gives

x=0,1=x^2+9y^2

y\'=0 gives

y=0,9=x^2+9y^2

9=x^2+9y^2 and 1=x^2+9y^2

are not both simultaneously possible

So critical points are

(0,0)

x=0 and 9=x^2-9y^2=0 ie x=0 and y^2=1

So, (0,1),(0,-1)

And , y=0 and 1=x^2+9y^2 ie x=1, -1

So,(1,0),(-1,0)

So 5 critical points