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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

autonomous system

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

thank you

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