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HW12 - Due 04/30/2008
ODE - spring 2008
This HW will count as 1/3 of the nal grade.
1) Study the stability of the critical points of x = 1 2x + x2 where is a parameter.
2) Consider x + sin(x) = 0 with x(0) = a and x (0) = 0 and 0 < a < .
a/ P
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HW11 - Due 04/23/2008
ODE - spring 2008
This HW will count as 1/3 of the nal grade.
1) Solve x +
1
4t2 x
= 0 with x(1) = 1 and x (1) = 0.
2) Consider the system
x = Ax + f (t, x) + g (t)
where A is a constant matrix, f and g are continuous an
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HW10 - Due 04/16/2008
ODE - spring 2008
1) For
> 0, approximate the solution of
x + x x3 = 0
with x(0) = 1, x (0) = 0 till the order
2
on a xed time interval. (The dierence is a O( 3 ).
2)Take
x = x x2 xy
y = y y 2 xy
(1)
1/Starting with posi
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HW8 - Due 04/09/2008
ODE - spring 2008
1) Consider the system
x = y + f (x, y )
y = sin(x)
(1)
where the function f is smooth.
Give some sucient condition on f so that (0, 0) is a stable equilibrium
2) Consider
x + (t)x = 0
(2)
with (t) C 1 (
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HW8 - Due 04/02/2008
ODE - spring 2008
1)What are the stability properties of the (0, 0) solution of
x = x + y n
y = y xn
(1)
depending on the parameters R and n N.
2)Show by an example that is f is C 1 and f (0) = 0, it is possible that limx
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HW7 - Due 03/26/2008
ODE - spring 2008
1)If 1 , 2 , ., n is a fundamental set for the homogeneous equation
L(x) = x(n) + a1 x(n1) + . + an x = 0
(1)
where a1 , a2 , ., an C (I ) are continuous functions of t, then nd the solution of L = b(t)
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HW6 - Due 03/12/2008
ODE - spring 2008
1) Find x such that
x(3) x(2) + 4x 4x = 0
(2)
and x(0) = x (0) = x
(1)
= 1.
2) What is the smallest n > 0 for which there is a dierential equation
x(n) + a1 x(n1) + . + an x = 0
having among its solution
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HW5 - Due 03/05/2008
ODE - spring 2008
1) Prove that if C is a real non-singular n by n materix, then there exists a REAL matrix A such that eA = C 2
2) If is a solution of x + a(t)x + b(t)x = 0 such that does not vanish on an interval I . Fi
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HW4 - Due 02/27/2008
ODE - spring 2008
1) By using polar coordinates, prove that the system
y = y + x y (x2 + y 2 )
x = x y x(x2 + y 2 )
(1)
has a unique periodic solution.
2) Consider the following system in R2 written in polar coordinates
r
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HW1 - Due 02/20/2008
ODE - spring 2008
1) For the following two dimensional system in R2
x = y (1 + x y 2 )
y = x(1 + y x2 )
(1)
determine the critical points and characterise the linearised ow in a neighbourhood of the these points.
2) What
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HW1 - Due 02/13/2008
ODE - spring 2008
1) Find the trajectory in the (x, x ) plan of the following equations
1. x x = 0
2. x + sin(x) = 0
2) Solve the following equations
1. 2t2 xx + x2 = 2
2. 2tx + t2 + tx x = 0
3) What is the domain of exis