Tow questions ( differential equation )

[little_lolo]

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

شكراً كتير مقدماً ​

 

[little_lolo]

please help me and i well be thanking for you

 

[HallsofIvy]

Please start by reading the files you were supposed to have read when you registered for this forum! You will not get any "help" if you refuse to even TRY doing the problem yourself!