There will be a quiz roughly every few weeks.
The material covered in each quiz will be described in advance on this web page, sometimes very explicitly. Many quiz questions will cover basic material that you should know absolutely perfectly.
Quiz 1: Friday 31 January.
You should be able to sketch the direction field for the
first order differential equation y'=ky, both when k>0 and when k<0.
You should also be able to find the general solution of this equation
(that is, show that every solution has
the form y=Cekx for some constant C).
We did this in class on the first day.
Solutions.
Quiz 2: Friday 21 February.
This will be a substitution problem, based on Section 1.6 methods.
It might be a type of substitution you have seen before (linear,
Bernoulli, homogeneous, or one of the homework types such as
v=x2+y2),
or it might be new, requiring you to make a reasonable guess.
Solutions.
Quiz 3: Friday 21 March.
Topic: Solution of second order linear
constant coefficient equations that arise in unforced mechanical
vibrations (Section 3.4).
Be able to determine whether the equation is overdamped,
critically damped or underdamped. In the underdamped case, you should be
able to solve an initial value problem and put the solution in the form
x=Ce -pt cos(wt-g), and be able to evaluate the amplitude C and
the phase shift g like we did in an example in class. (Take another look
at that example!) Practice sketching graphs of the solutions.
Solutions.
Quiz 4: Friday 25 April.
Topic: Fourier series. The problem will
be similar to one of the Homework 9 problems on Fourier series.
You will also have to sketch the graph of the function,
showing it over several periods.
Remember that the Fourier series of f equals f(t)
except at a jump point. At a jump point the series equals the average
[f(t+)+f(t-)]/2 of the values of f from the right and from the left.
Solutions.
Quiz 5: Monday 5 May.
Topic: Theorem 2 in Section 9.5.
(Recall that the phrase ``insulated ends'' means Neumann boundary conditions.)
You will be asked to derive this result by using Separation of Variables.
Follow the derivation on pages 638-639, but note the following points...
You must include the steps (16)-(20) in your solution.
You should not
include proof of the fact that the eigenvalues of X'' + lambda X = 0 on
0 < x < L under Neumann boundary conditions are
lambdan=(n pi/L)2 for n=0,1,2,..., with
corresponding eigenfunctions Xn(x)=cos(n pi x/L). This fact was on
Test 2, and so you can just state it and use it without proof, on the Quiz.
Note: in the special case n=0, the eigenvalue and eigenfunction are just
lambda0=0 and X0(x)=1.
Solutions.