MA 1270:
EXAM 1 Review
Problems similar to the following will be given in the exam.
- Modeling problems. These include one or two-dimensional
differential equation models where you will write down a differential
equation for a given phenomena. Chapter 1.1 and 2.1 are the relevant
ones to look at. For example, I might ask you to model a two-species
population which has the following behavior:
- Both species die out in absence of each other
- Each species helps the other grow in proportion to the rate at
which they interact
- The first species growth rate is limited by crowding.
- Separation of variables. Know how to solve anything of the form
dx/dt = f(x)g(t)
For example, solve
dx/dt = (x^2 + 1)t
x(0)=1
- Slope fields. You will have to match the slope fields of a bunch of equations
with the corresponding differential equation such as problem 11 on page
48.
- Phase-line. You should know how to draw the equilibria and phase-line for
first order differential equations like exercises 1-8 and 9-15 on page
89. You should also know how to do problems like 27-30.
- Eulers method. You should be able to give a few approximations to Eulers method
for a given differential equation. Recall that
x(n+1) = x(n) + h f(x(n),t(n))
- Bifurcations. You should know a little bit about
bifurcations. You should know that a bifurcation occurs when the
number of equilibrium points changes as a parameter varies. Here is an
example. Find the bifurcation point(s) for the differential equation
and sketch the bifurcation curve.
dx/dt = x exp(-x) - mu mu>0
- Nth order ODEs/stability.
You should know how to find the general solution to any nth order
homogeneous constant coefficient differential equation and also the
Routh-Hurwitz criterion. Here are some examples:
- Solve D^2(D^2+4)^2y = 0
- For what values of the parameter p do all solutions to
x''' + p^2 x'' + x' + (1-p)x =0
decay exponentially?
- Linear 1st order ODEs.
You should be able to solve problems of the form
dx/dt = a(t) x +b(t).
such as
dx/dt = 2 x/t + t^2, x(1)=4
Selected hints
Let X,Y be the two
species. Then a possible model is
dX/dt = -k1 X + k2(1-X/N)X Y
dY/dt = -k3 Y + k4 X Y
The first terms in each equation are the death rates. The second terms
in each equation are the growth due to the interaction. Note that
species X has a growth rate than hits zero when X=N and is negative if
X>N due to growth limited crowding.
dx/(x^2 + 1) = t dt
Integration yields
atan(x) = (t^2)/2 + C
The initial condition implies that C=pi/4 since
atan(1)=pi/4. Thus
x(t) = tan((t^2)/2 + pi/4)
Recall that a
bifurcation occurs if f(x,mu)=0 and df(x,mu)/dx=0. So
df(x,mu)/dx = exp(-x) - x exp(-x) = 0 implies
(1-x) exp(-x) = 0 so
x=1
Thus the critical value of x is 1 and since f(1,mu)=0, this means,
mu=1 exp(-1) = 1/e. The diagram is
The characteristic polynomial is x^2(x^2+4)^2. x=0 is a double root
and x=2i, -2i are also double roots. So the general solution is
y(t)= A+Bt + (C+Dt)sin(2t)+(E+Ft)cos(2t)
The Routh Hurwitz criterion for third order equations implies that
p^2>0, 1-p > 0, p^2 -(1-p) > 0
Solving the quadratic inequality (finding the roots of the quadratic)
means that
(-1/2 + sqrt(5)/2) < p < 1
Recall that the general solution is
x(t) = exp(A(t)) [ C + int{ exp(-A(t)) b(t) dt}]
A(t) = int{a(t) dt}.
Thus, for the present problem, A(t) = 2 ln(t) = ln(t^2), so
x(t) = t^2 [ C + int{ (1/t^2) t^2 dt}] = C t^2 + t^3
The initial condition implies that C=3 so x(t)=3t^2+t^3.