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 $f(x) = \sqrt{x}$ 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 y₁ and y₂ are solutions to the above DE. Prove that for any real numbers c₁ and c₂, y=c₁y₁ + c₂y₂ 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:

$${\bf x}'=\left(\begin{array}{rrr} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{array}\right){\bf x}.$$

HINT: $\lambda^3 - 3\lambda - 2 = (\lambda - 2)(\lambda + 1)^2$.

4.

Find the general solution of the DE:

$$ty' + y = 3t\cos(2t) \qquad t>0.$$

5.

Consider the function f defined by

$$f(x) = \left\{\begin{array}{cl} x & \qquad 0 < x \leq 1\\ 2-x & \qquad 1 < x \leq 2 \end{array}\right.$$

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 a²u_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 v₀ 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:

$$yy' = x^2 \qquad y(0)=4$$

9.

Consider the linear homogeneous DE

$$y'' + \frac{1}{t}y' + t^2y = 0 \qquad t>0.$$

Suppose that y₁ and y₂ are two solutions the above DE. Compute the Wronskian of y₁ and y₂ (note that you need not know what y₁ and y₂ are in order to know their Wronskian).

10.

Find the general solution of the DE:

$$y'' - y' - 2y = 2\sinh(2t).$$