Practice Exam -3
Intro. DEs
Spring 2005
1. Consider the folling system of differential equations which is supposed to model two interacting species:
dx
= 5x − x2 − xy,
dt
dy
= −2y + xy
dt
(a) Would this system be modeling a coorperative, competetive, or
predator-prey situation ?. Exlpain.
(b) Find all equilibrium solutions.
(c) Determine the type and stability of the critical points.
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2. Assume that the following system is almost linear:
dy
dx
= f (x, y),
= g(x, y)
dt
dt
Let (0, 0) be a critical point of the system. Prove
dx
dy
= fx (0, 0)x + fy (0, 0)y,
= gx (0, 0)x + gy (0, 0)y
dt
dt
is the linearized system of the almost linear system at (0, 0).
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3. Consider the linear system
dy
dx
= ǫx − y,
= x + ǫy
dt
dt
(a) Determine the type and stability of the critical point (0, 0) if
(i) ǫ < 0
(ii) ǫ = 0
(iii) ǫ > 0
(b) Explain why small perturbation of the system change the type
and stability of the critical point.
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4. Let f (t) be continuously differentiable function such that
|f (t)| ≤ Mect
∀t.
Denote F = L{f (t)}. Prove the followings:
(a) L{f ′ (t)} = sF (s) − f (0) for any s > c.
(b) L{eat f (t)} = F (s − a) for any s > a + c.
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5. Using Laplace Transform Method, solve the initial value problem
(a)
x′′ + 4x = cos 3t,
x′ (0) = x(0) = 0.
(b)
y ′′ + 4y ′ + 4y = 8t2 ,
y(0) = y ′ (0) = 0
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6. Using Laplace Transform Method, find the solution of the system
x′′ = −3x + 2y,
y ′′ = 2x − 2y + 40 cos 3t
subject to the initial conditions
x(0) = x′ (0) = y(0) = y ′ (0) = 0.
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L{tn } =
n!
, s>0
sn+1
1
L{eat } =
, s > a,
s−a
s
L{cos kt} = 2
, s>0
s + k2
k
, s>0
L{sin kt} = 2
s + k2
s
L{cosh kt} = 2
, s > |k|
s − k2
k
L{sinh kt} = 2
, s > |k|
s − k2