# Math 267, Section 203 Problems, Solutions, Handouts

Note: PDF files may be read with Acrobat Reader, which is available for free from Adobe.

### Notes

• Review of Ordinary Differential Equations [ pdf ]
• The RLC Circuit [ pdf ]
• Derivation of the Wave Equation [ pdf ]
• Derivation of the Telegraph Equation [ pdf ]
• Solution of the Wave Equation by Separation of Variables [ pdf ]
• Fourier Series [ short version , long version (version of Feb 4, 2007) ]
• Solution of the Heat Equation by Separation of Variables [ pdf ]
• Periodic Extensions [ pdf ]
• Complex Numbers and Exponentials [ pdf ]
• The Fourier Transform [ pdf ] (version of Feb 25, 2007)
• Using the Fourier Transform to Solve PDE's [ pdf ]
• Discrete-Time Fourier Series and Transforms [ pdf ] (version of Mar 21, 2007)
• Discrete-Time Linear Time Invariant Systems and z-Transforms [ pdf ] (version of Apr 4, 2007)

### Supplementary Material

• Kermit Sigmon's MATLAB Primer [ pdf ]
• MATLAB Practice Lab [ pdf ]
• MATLAB script, named waveeqn.m, which creates an animation illustrating the behaviour of solutions to the wave equation. The initial position is a single spike. The initial velocity is zero.
• MATLAB script, named heateqn.m, which creates an animation illustrating the behaviour of solutions to the heat equation. The initial position is a single spike.
• Derivation of the Heat Equation for One Space Dimension [ pdf ]
• Long Division [ pdf ]

### Applets

• Fourier Series. This demo shows a Fourier series adding up to the expected answer.
• The Wave Equation. This demonstration illustrates the behaviour of solutions of the wave equation. The demonstration plots the solution given by separation of variables that you have found in class. Separation of variables expresses the solution as a sum b_1(t) sin(pi x/l) + b_2(t) sin(2 pi x/l) + ... of modes. When the demonstration starts, the initial amplitude is plotted. By clicking the "Advance time" button, you instruct the computer to increase the time by an amount specified in the "time interval window".
• The Wave Equation (animated) [ one bump | two bumps ]. These two demonstrations also illustrate the behaviour of solutions of the wave equation. They also plot the solution given by separation of variables that you have found in class. The demonstrations animate the solution by successively plotting u(x,0), followed by u(x,dt), followed by u(x,2dt) and so on. The two demonstrations use different initial conditions.
• The Telegraph Equation. This program gives an animated demonstration of the solution to the telegraph equation. The viewer may adjust the equation parameters to give signal transmission with and without distortion due to dispersion.