Please can you solve this tow questions today...
Q1) If g is a function such that g(0)=0 and all high order derivatives exist consider the
dx/dt = g(y) dy/dt = g(x)
a. show that (o.o) is critical point and that system is almost linear in the neighborhood of (o.o)
b. show that if g'(o)>o then critical point (o,o) is unstable and that if g'(o)<o then the critical point is asymptotically stable
c. show that the critical point (o,o) is a saddle point and unstable
Q2) consider the system
dx/dt =f(y) dy/dt =g(x)
where f,g are functions whit all their higher derivatives exist and f(o)=g(o)=o and f'(0)≠0 g'(o)≠o
a. show that (o.o) is critical point of the system and the system is almost linear system at it.
b. show that if f'(0)g'(0)>0 then the critical point (0.0) is a saddle point and if f'(0)g'(0)<0 then the critical point (0.0) is a center or spiral point
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