Tow questions ( differential equation )



  1. 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
    :biggrin:
     
  2. jcsd
  3. please help me and i well be thanking for you :smile:
     
  4. HallsofIvy

    HallsofIvy 40,241
    Staff Emeritus
    Science Advisor

    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!
     
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