Math 308 - Differential Equations Fall 2002
Homework Assignment 8
Due Friday, November 8.
For Questions 1-3,
(a) Find the general solution to the system d~
Y
dt =A~
Y.
(b) Sketch the phase portrait. Use nullclines to help make reasonably accurate plots of the trajectories.
(c) Classify the equilibrium point of the system as either a spiral sink, a spiral source, or a center. (The
matrices have been chosen so that these are the only possibilities.)
(d) Solve the initial value problem ~
Y(0) = ·2
1¸. Draw this solution in your phase portrait in part (b).
(Use a diferent color or use a dashed line to indicate this solution.) Also sketch x(t) and y(t) (where
~
Y=·x
y¸) for this solution, including positive and negative values for t.
1. A=·2/3 4
−4 2/3¸
2. A=·0 2
−2−2¸
3. A=·−2 3
−3 2¸
4. Let A=·a b
−b a¸, where b6= 0. Show that Amust have complex eigenvalues.
5. Recall the second order differential equation for the mass-spring system:
md2y
dt2+bdy
dt +ky = 0,
where m > 0, b≥0, and k > 0.
(a) Let v=dy
dt , and convert this equation into a 2 ×2 first order system for ~
Y(t) = ·y(t)
v(t)¸.
(b) What conditions on m,b, and kwill ensure that ~
Y(t)→~
0 as t→ ∞?
(c) Under what conditions on m,b, and kwill the solutions exhibit a decaying oscillation?
For Questions 6-8,
(a) Find the general solution to the system d~
Y
dt =A~
Y.
(b) Sketch the phase portrait.
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