Find and classify the critical point of the plane autonomous

Find and classify the critical point of the plane autonomous system x\' = 2 + x/y y\' = x/y - x

Solution

Conside autonomus system of DE

x\'=P(x,y)

y\'=Q(x,y)

this is called plane autonomus system

the constant solution X(t)=(x,y) for all t

P(x,y)=0 and Q(x,y)=0

then (x,Y) called critical point

given plane autonomus system is

x\'=2+x/y

y\'=x/y -x

for critical point

consider

2+x/y=0

and x/y -x=0

that is

2y+x=0-----------------(1)

and x-xy=0-------------------(2)

multipling (1) by y and add in (2) we get

2y²+x=0------------------(3)

now substract (1)from(3) we get

2y²-2y =0

2y(y-1)=0

that is y=0 or y-1=0

but y is never 0 because if y=0 than 2+x/y is not define

therfore y-1=0

that is y=1

put y=1 in (1) we get x=-2

therfore (x,y)=(-2,1)

is critical point for given plane autonomus system