# Practice Exam - 3 Intro. DEs Spring 2005

```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.
1
2
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).
3
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 &lt; 0
(ii) ǫ = 0
(iii) ǫ &gt; 0
(b) Explain why small perturbation of the system change the type
and stability of the critical point.
4
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 &gt; c.
(b) L{eat f (t)} = F (s − a) for any s &gt; a + c.
5
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
6
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.
7
L{tn } =
n!
, s&gt;0
sn+1
1
L{eat } =
, s &gt; a,
s−a
s
L{cos kt} = 2
, s&gt;0
s + k2
k
, s&gt;0
L{sin kt} = 2
s + k2
s
L{cosh kt} = 2
, s &gt; |k|
s − k2
k
L{sinh kt} = 2
, s &gt; |k|
s − k2
```