Partial Differential Equations

Suppose that a string of length L is held fixed at one end and is being moved up and down, say with a displacement of f(t), at the other end. Assume that the string is initially at reast and has a zero displacement, and similarly, that the forcing of the string, f(t), also has a zero initial velocity and displacement. a) Write down the system of equations that models this physical problem. b) Think of some change of dependant variables that produces another wave equation problem with homogeneous boundary conditions. (The other equations may become inhomogeneous as a result.) c) Solve the equations from part (b) using the method of eigenfunction expansions. d) Describe what happen... click for more

Electrostatic Potential

In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... Please see attached... Let us assume that the potential is known at z = 0, so that V(r, theta, z)= f(r, theta) is a given function. Because of the symmetry in the problem, V(r, theta, -z) = V(r, theta, z), so we only have to consider the semi-infinite cylinder in the domain z>0. Since the cylinder is grounded, the potential there must be zero, meaning that V(1, theta, z) = 0... Please see attached for full... click for more

Separation of variables - turkey

You are trying to cook a turkey for Thanksgiving, but the instructions on the turkey are for a 15-pound turkey and you have a 20-pound turkey. In this problem, you will use separation of variables to estimate how long it should take to cook your 2-pound turkey.

ODEs: Sturm-Liouville Form; Weight Function; Inner Product

Write each of these ODEs in Sturm-Liouville form, and identify the weight function that woudl be used in defining an appropriate inner product {see attachment for ODEs}

num solutions to PDE that approach zero as x -> (+/-)infinity

Does y' = y - e^(-(x^2)) have solutions that approach zero as x goes to infinity and negative infinity? how many?