MA501 Practice Test # 1 January 30, 2017 Please, show your work. This way I might be able to give you a partial credit even if the result is wrong. 1. (40 pt.) Consider linear differential equation x2 y + xy 4y = x2 + x, x>0 a) find the general solution of the associated linear homogeneous equation. c) find the general solution of the original nonhomogeneous differential equation. 2. (40 pt.) Verify whether x = 0 is a regular singular point of the equation x2 y + x(x 2)y + (x2 + 2)y = 0 Use Frobenius method to find a solution of this differential equation around x = 0. Find the indicial equation, the recursive relation for the first solution and the first four terms for this solution. 3. (20 pt.) For the function f (x) given on the interval [2, 2] as follows: f (x) = x, 2 x < f (x) = sin x, x < f (x) = x + , x 2 verify whether its Fourier series converges on the interval [2, 2] and compute the sum of the Fourier series at the following values of x: x = 2, x = , x = , x = , x = 2 2