Math 5110/6830
Instructor: Alla Borisyuk
Homework 5.1
Due: October 17
1. Analyze the following equations graphically. In each case sketch the
vector eld on the real line, nd all the xed points, classify their stability, an
sketch the graph of ( ) for dierent initial conditions. Then try to solve the
equation analytically. It may not be possible! - state it if you think so.
3
a) _ =
x
b) _ =
sin
x t
x
x
x
e
x
x
2. Suggest an equation that is consistent with the following phase portrait:
0
−1
2
3. Consider a bacterial population whose growth rate is
dN=dt
. Show that
()=
N t
N0
= ()
K t N
exp
Z
t
()
K s ds
0
:
4. Solve the logistic equation _ = (1
) analytically for an arbitrary
initial condition 0 . (Hint: Separate variables and integrate, using partial
fractions OR make the change of variables = 1 ).
N
rN
N=K
N
x
=N
5. Cancerous tumor growth can be modeled by the Gompertz law:
_=
ln( )
N
aN
bN ;
where ( ) is proportional to the number of cells in the tumor and
0
are parameters. Sketch the vector eld and then graph ( ) for various initial
values.
N t
a; b >
N T
1