Math 5C 2005
Practice Midterm

1. Consider the equation
   
   $$\;x^2 y''+x y'+ (x^2-1) y=0$$
   
   
   1. Show that $x_0=0$ is a regular singular point.
   2. Find the indicial equation for the regular singular point $x_0=0$.
   3. For the largest value of solution of the indicial equation, find the general form of the series expansion of the solution.
2. Given the function .
   1. Find the Fourier sine series expansion of $f(x)$.
      
   2. Graph the function to which the Fourier series in part (a) converges over the interval $[-1,1]$.
   3. Find the Fourier cosine series expansion of $f(x)$.
      
   4. Graph the function to which the Fourier series in part (c) converges over the interval $[-1,1]$.
3. Compute the Fourier series of the function
   
   $$f(x)={1\over 2}\sin (6 \pi x) - {3\over 5} \cos (14\pi x) + \frac{9}{8}\cos(24\pi x)$$
   
   

4. Solve the differential equation
   
   
   
   using a Fourier series.