Oscillations and Waves: Winter-2026
HW 1b: Due W1 D5

1. Fourier Series of a Triangle Wave

   Consider the following triangle wave:

   

   1. Find the Fourier series for a triangle wave (such as the one shown in the figure), which has amplitude \(A\) and period \(T\).
   2. Plot two approximations to your solution, one including the first nonzero term and the other including the first four nonzero terms.
   3. Make a histogram of your coefficients, i.e. find the spectrum.

2. Homogeneous Linear ODE's with Constant Coefficients

   Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:

   Constant Coefficients, Homogeneous

   or your differential equations text.

   Answer the following questions for each differential equation below:

   • identify the order of the equation,
   • find the number of linearly independent solutions,
   • find an appropriate set of linearly independent solutions, and
   • find the general solution.
   Each equation has different notations so that you can become familiar with some common notations.
   1. \(\ddot{x}-\dot{x}-6x=0\)
   2. \(y^{\prime\prime\prime}-3y^{\prime\prime}+3y^{\prime}-y=0\)
   3. \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)