ME760 Engineering Analysis I

Homework Set 7 due: Monday Nov. 23, 2015 1. Find two power series solutions about x = 0 of the differential equation (1 - x2)y00 - 3xy0 + y = 0. Show that when the value of = N(N + 2) (N an integer) the corresponding power series becomes an Nth degree polynomial UN(x). Construct U2(x) and U3(x). 2. Verify that x = 0 is a regular singular point of the equation x2y00 - 3 2 xy0 + (1 + x)y = 0, and that the indicial equation has roots 2 and 1/2. Show that the general solution is y(x) = 6aox2 1X n=0 (-1)n(n + 1)22nxn (2n + 3)! +bo x1/2 + 2x3/2 - x1/2 4 1X n=2 (-1)n22nxn n(n - 1)(2n - 3)! ! . 3. Use the derivative method to obtain as a second solution of Bessel’s equation for the case when = 0 the following expression. y(x) = Jo(x) ln x - 1X n=0 (-1)n (n!)2 Xn m=1 1 m !x 2 2n . 4. Find the general series solution about x = 0 of the equation x d2y dx2 + (2x - 3) dy dx + 4 x y = 0.