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**Unformatted text preview: **, f/ufl‘OYLW Instructor:H. Zhu April 1, 2009
MATH 2270: Differential Equations TEST I Last Name: __—First NaIne:*_ ID: 1. E Find the general solution in vector form and solve the initial value problem: 2m + 3ty PM [JULLAj lbw/Old 2ty
110(0) 2 1, y(0) = 1. . Ha
(e\ 8\
H II .30 By changing variable(s), write the following equations into either separable or linear equations. If it is a linear equation, ﬁnd one integrating factor. You do NOT
need to ﬁnd the general solution/\_,.—-—— (mﬁ— 562+? —, / kit-Yr:
W— $2_y2- \/ H505
b) COS’LUd—w = t Sinw+ 1 1-H.
( I d 1+t2 14-752.
(gﬁz y .
d2: $+y3
1 ﬂ / /
417 du
:Z “PU
‘TX W
A ”’f M
U
2cm +u:/ H};
it}. _. 0“
2 O“ “ +153? “U
+\ 3. IE Consider the ﬁsh population model with harvesting: dP 1 P 8
5:31)“??-
(a) Find the equilibrium solutions of the model.
(b) Classify each of these equilibrium solutions as sink or source.
(c) Plot the phase line.
(d) Find the general solution. (e) Discussion: given the initial ﬁsh population size P(0) = P0 > 0. What do these
results imply about the long term behavior of the ﬁsh population? Answer this
question for the case when P0 = 1, P0 = 6 and P0 = 12 and plot the solutions. (f) Consider the model with harvesting dP 1 P
—=— 1———h
dt 3“ 10) where h > 0 is a parameter. Locate the bifurcation value for the one—parameter
family and draw the phase lines of parameter slightly smaller than, slightly larger
then, and at the bifurcation value. @- hf ELK/49460 fzp ...

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