**DISCLAIMER:** While I would consider this an appropriate final exam, it should in no way
be interpreted as a comprehensive list of topics to study for the actual final exam. We covered
far too many topics in this course for them all to appear on any single 2-hour exam.
Thus there are likely to be topics covered on the final which do not appear here and there are
likely to be topics covered here which will not appear on the final. I could go on disclaiming
for pages, but I hope you get the idea.

**NOTE:** Time-permitting, I will post solutions and/or hints. Keep your eye on the main math
23 web page. Also, I typed this up rather hastily, so there may be typos. Please let me know if
you find any.

**NOTE #2:** I'm not sure about this, but it is possible that this practice final might be a
bit long.

- 1.
- (a)
- Compute the Taylor series for
about
*x*=1. - (b)
- What is the interval of convergence of the series you found above? Justify your
answer.

- 2.
- Consider the linear homogeneous DE

*y*'' +*p*(*t*)*y*' +*q*(*t*)*y*= 0.

(assume*p*and*q*are continuous everywhere). Suppose that*y*_{1}and*y*_{2}are solutions to the above DE. Prove that for any real numbers*c*_{1}and*c*_{2},*y*=*c*_{1}*y*_{1}+*c*_{2}*y*_{2}is also a solution to the same DE. Note that this*is*a theorem we proved in class. You are therefore not allowed to merely quote it. - 3.
- Solve the system of equations:

HINT: .

- 4.
- Find the general solution of the DE:

- 5.
- Consider the function
*f*defined by

Find a Fourier sine series for

*f*. - 6.
- Suppose you have an elastic string of length 2 cm, and suppose that the ends of that string
are held fixed in such a way that
*a*=1 (in the general wave equation*a*^{2}*u*_{xx}=*u*_{tt}). Now suppose you configure that string into the shape of the graph of the function*f*from the previous problem, and then you release it (i.e. no initial velocity). Find a function*u*(*x*,*t*) which describes the displacement of the string at location*x*at time*t*. - 7.
- Suppose a mass of
*m*kg is attached to a spring with spring constant*k*. Assume there is no air resistance and assume there is no external force acting on the mass. Assume that the mass is resting in its equilibrium position and, at time*t*=0, it is flicked upward with an initial velocity of*v*_{0}m/s. Find a function*u*(*t*) which describes the position of the mass at any time*t*>0. - 8.
- Solve the initial value problem:

- 9.
- Consider the linear homogeneous DE

Suppose that*y*_{1}and*y*_{2}are two solutions the above DE. Compute the Wronskian of*y*_{1}and*y*_{2}(note that you need not know what*y*_{1}and*y*_{2}are in order to know their Wronskian). - 10.
- Find the general solution of the DE: