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Unformatted text preview: x = x-ωy ˙ y = ωx + y with the initial condition x (0 ,ω ) = 1, y (0 ,ω ) = 0. Is the solution a smooth function of ω ? Let X = ∂x/∂ω and Y = ∂y/∂ω . What equation, with what initial condition, does the pair ( X,Y ) satisfy? 2 9. Consider the solution ( x ( t,x ) ,y ( t,y ) of the problem ˙ x = x-y ˙ y = x + y with the initial condition x (0 ,x ) = x , y (0 ,y ) = y . Is the solution a smooth function of ( x ,y )? Let X = ∂x/∂x and Y = ∂y/∂y . What equation, with what initial condition, does the pair ( X,Y ) satisfy? 10. Let X : U ⊂ R n → R n be a smooth vector ﬁeld on an open set containing the origin. Suppose that X (0) = 0. Let T > 0 be a given positive real number. Show that there is a ball B about the origin such that any initial condition x ∈ B has a positive lifetime that is at least T ....
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This note was uploaded on 02/23/2010 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
- Fall '09