Find all critical points of the given plane autonomous syste

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

x\' = x(1 x^2 5y^2)

y\' = y(5 x^2 5y^2)

Solution

To find the critical points, we need to find every ordered pair of real numbers (x, y) at which both x and y are zero.

This means algebraically solving the system

0 = x( 1 - x2 - 5y2 )

0 = y( 5 - x2 - 5y2 )

Fortunately, the first equation factors easily

and gives x=0

while from 2nd eqn we get

y=0

If x = 0, then the second equation in system reduces to

y( 5 - 5y2 )=0

y = 0 , 1 , -1

so,the critical points will be

(0,0)

(0,1)

(0,-1)

If y = 0

then the first eqn reduces to

x( 1 - x2 ) = 0

x = 0 , -1 , 1

so,the critical points are

(0,0)

(1,0)

(-1,0)