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homework4

homework4 - x = x-ωy ˙ y = ωx y with the initial...

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1 CDS 140a: Homework Set 4 Due: Friday, October 30st, 2009. 1. Show that if A is diagonalizable, then det e A = e trace A Try this out on a few nondiagonalizble matrices and make a conjecture as to its general validity. 2. Solve the system ˙ x = ax - by ˙ y = bx + ay by using polar coordinates. 3. Find the stable, unstable and center subspaces for the system ˙ x = y ˙ y = 0 ˙ z = 2 x - z and comment on the phase portrait. 4. Find the stable, unstable and center subspaces for the system ˙ x = y ˙ y = - x ˙ z = 2 x + z and comment on the phase portrait. 5. Does the one dimensional equation ˙ x = x 1 / 3 , x (0) = 0 have a unique solution x ( t ) deﬁned for t in some interval ( - ±,± )? 6. What happens when you apply Picard iteration to the linear system ˙ x = Ax ? 7. Use the local existence and uniqueness theorem to estimate the time of exis- tence of the solution of the one dimensional equation ˙ x = x 2 , where x (0) = 1. What is the actual positive lifetime of the solution? 8. Consider the solution ( x ( t,ω ) ,y ( t,ω ) of the problem ˙

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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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