Convert this system into polar form and find the polar

Question

Phase paths in polar form Show that the system of equations

x^·₁ = F₁(x₁,x₂,t),      x^·2 = F₂(x₁,x₂,t)

can be written in polar coordinates in the form

r^· = (x₁F₁ + x₂F₂) / r,

θ^· = (x₁F₂ - x₂F₁)/ r².

where x₁ = r cos θ and x₂ = r sin θ. A dynamical system satisfies the equations

x = -x + y,
y = -x - y.

Convert this system into polar form and find the polar equations of the phase paths. Show that every phase path encircles the origin infinitely many times in the clockwise direction. Show further that every phase path terminates at the origin. Sketch the phase diagram.