Homework 6
For the following three problems, rst solve the equation ( ) = 0 to nd
the critical points of the given autonomous dierential equation
= ( ).
Then analyse the sign of ( ) to determine whether each critical point is stable
or unstable, and construct the corresponding phase diagram for the dierential
equation. Next solve the dierential equation explicitly for ( ) in terms of .
f x
dx=dt
f x
f x
x t
1.
2.
3.
4.
x
x
x
0 = x2
0=9
t
4
x
2
x
0 = (x 1)3
Consider the dierential equation
= +
dx
x
dt
3
kx
containing the parameter . Analye the dependence of the number and
nature of the critical points on the value of , and construct the corresponding bifurcation diagram.
5. Seperate variables in the logistic harvesting equation
k
k
dx
dt
= (
)(
k N
x
x
H
)
and then use partial fractions to derive the solution
()= (
(
x t
N x0
H
x0
H
) k(N H )t
)
(
) ( 0
H x0
N e
) k(N H )t
x
N e
:
6. Consider the two dierential equations
dx
dt
and
dx
dt
=(
a
=(
x
x
)(
)(
a
x
b
)(
b
)(
x
x
c
)
c
)
x ;
each having the critical points , , and . Suppose that
.
For one of these equations only the critical point = is stable, for the
other equation = is the only unstable critical point. Construct phase
diagrams forthe two equations to determine which is which. Without
attempting to solve either equation explicitly, make rough sketches of the
typical solution curves for each. You should see two funnels and a spout
in one case and two spouts and a funnel in the other.
a
b
c
a < b < c
x
x
b
1
b