## Transcribed Text

2. Consiler the matrix A - (i )) For and 1, classify equilibiom point the
e - 1 - the at. origin
for the - Aur. Bonns problem (not gratel): do thesame for all values of c.
3. Consider the equation z' - where A-
(a) For c = 1 and c = 1, classify the equilibrium point at the origin.
(b) Find a value of c so that the origin is an unstable node.
Bonus problem (not graded): classify the critical point for all values of c.
1. Consider the differential equation y" + by + cy - 0 where 0 and c are constants.
(a) Turn this equation into a system of first order linear ODEs.
(b) Find the characteristic polynomial of the coefficient matrix you found in part (a). llow does this
polynomial relate to the original second order equation?
a. Consider the system 2((0) - M(t).
(a) Tal. 60 be a fixed sulution of this equation. Tf is any sulution LA the inhomogemoos
equation, then slow that g(i) - + where is sume solution to the homogemeous
system F'(t) - A(t)F(t). (Ilint: consider the difference 5(t) -
(b) Conversely, show that it 5(t) is any solution to the bomogeneous equation, then -
is a solution to the exquation.
(e) Interpret parts (a) and (b) to conclude that the general solution of the inhomogeneous equa-
tion ermais the general solution of the homogeneons equitation plus 8 partionlar solution of the
equation.

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