## 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

#### Modeling problem.

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.

#### Separation of variables.

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)

#### Bifurcation problem.

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 #### Nth order ODEs

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

#### Linear 1st order ODEs

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.