The second exam covers Chapter 3 and 4 and Sections 5.1 and 5.2 of the course text. Note that we only skimmed Sections 3.3 and 3.4 and did not cover Section 3.5. It is highly recommended that you go over homework problems (solutions are on reserve in the black "Roberts" binder in the Math/Sci. Library) and your class notes.

**Note:** You will be allowed one "cheat sheet," a single 8.5 x 11
piece of paper, front and back, full of whatever formulas, graphs, etc. you wish.

**Exam Review:** We will review for the exam on Monday, April 11th from 6:00 - 7:30 pm
in Swords 359. Please come prepared with specific questions.

The following concepts, definitions and equations are important material
for the exam:

- Diffusion and Waves on the Half-line 0 < x < infty: odd and even extensions, how to solve a given equation with either Dirichlet or Neumann boundary conditions, unwinding D'Alembert's formula or the formula for the solution to the diffusion equation
- Waves with a Source: Using the u_h + u_p technique to solve u_tt - c^2 u_xx = f(x,t) and solving for the arbitrary functions given the initial conditions
- Boundary Problems: Separation of variables technique, solving PDE's with different boundary conditions (Dirichlet, Neumann, Periodic, Robin) as infinite series, finding eigenvalues and eigenfunctions, frequencies, using graphical analysis to find eigenvalues
- Fourier Series: Finding coefficients for the Fourier sine series, Fourier cosine series,
full Fourier series, and complex full Fourier series; even, odd and periodic functions and
how their properties simplify finding Fourier coefficients
**Important Items from Exam #1** - General PDE terminology: operators, linear operators, homogeneous and inhomogeneous linear equations, well-posed problems (existence, uniqueness and stability)
- Initial and Boundary Conditions: physical interpretations of initial conditions (position, velocity, etc.), describing conditions mathematically, boundary conditions (Dirichlet, Neumann, Robin), physical interpretations of boundary conditions
- The Wave Equation: general solution, D'Alembert's formula
- The Diffusion Equation: physical interpretation of solutions, general solution, the source function S(x,t), the error function Erf(x)

**Some Practice Problems:**

**Note: These are practice problems useful for reviewing the material. Many are
too lengthy for a timed exam but parts of these problems make suitable exam
questions.**

** Chapter 3 **

Section 3.1 : 2

Section 3.2 : 2

Section 3.4 : 1

**Hint:** For problem #2 in Section 3.1, first let v(x,t) = u(x,t) - 1 and find
the PDE and boundary conditions that v(x,t) satisfies. For problem #2 in Section 3.2,
use the solution derived in problem #1 of the same section. This problem is very reminiscent
of problem #5 in Section 2.1.

**Chapter 4 **

Section 4.1 : 2, 4

Section 4.3 : 6, 9

**Chapter 5 **

Section 5.1 : 5, 6

Section 5.2 : 1, 3, 4, 8