MTH 3201 Midterm Test 1: Spring 2010: Solutions
1
First Problem
Solve the heat equation ut = 9uxx for a rod of length with both ends held at zero temperature (zero Dirichlet
boundary conditions) if the initial temperature is given by (x) = x( x). First, f
MTH 3201 Final Exam-E1: Fall 2010: Solutions
1
1
Vibrations in a Rectangular Membrane
Consider the vibrations of the rectangular membrane with two edges fixed and the other two edges
held free.
a. (25 points) Find the general formula for the displacement
MTH 3201 Final Exam-01: Fall 2013
1
Name:
Vibrations in a Rectangular Membrane
Consider the vibrations of the rectangular membrane with the edges held free.
a. (25 points) Find the general formula for the displacement u(x, y, t) of the membrane by solving
MTH 3201 Final Exam-01: Fall 2010: Solutions
1
1
Heat Conduction in a Cylinder
Consider Fourier heat conduction in a cylindrical body with a rectangular cross-section. Assume
that the temperature is independent of the cylindrical axis, two of the plane la
Practice Final Exam - MTH 3210 - Intro to PDE & Apps - Dr.
Kanishka Perera - Spring 2014
1. (a) (65 points) Solve
ut = u, 0 < x < a, 0 < y < b, t > 0
u(x, 0, t) = ux (a, y, t) = u(x, b, t) = ux (0, y, t) = 0
u(x, y, 0) = f (x, y).
(b) (5 points) Find u(x,
Homework 13
1
August 8, 2012
9.3
1. a. Calculate in cartesian and polar coordinates that xy = 0.
b.
(2x3 + 3x2 y y 2 ) = 12x + 6y 2
The calculation in polar coordinates, while tedious, yields the same result.
2. a.
y 2 (x2 + y 2 )4 = 14r2 sin2 + 2r2 cos2
Homework 13
1
November 15, 2010
9.3
1
a. Calculate in cartesian an polar coordinates that xy = 0. In cartesian coordinates, we
have
2 xy 2 xy
+
x2
y 2
y x
+
=
x y
=0
xy =
In polar coordinates, we want to show that [r2 sin cos ] = 0. But note that we can