Find a general solution in powers of x of the differential equation. State the recurrence relation and the guaranteed radius of convergence. (x²-9) y" + 4xy' + 2y = 0 The recurrence relation is Cn +2= Find a general solution in powers of x. y(x) = (Type an expression in terms of co and c₁. Type any series in summation notation using n as the index variable and as the starting index.) Find the guaranteed radius of convergence p. p=

Assume the solution of = 5 – y, y(0) = 2 can be written in the form of a power series, y = E-o Cnx". dt Determine the first three nonzero coefficients co, C1, and c2. Then you have a second-degree polynomial that approximates the true solution of the initial value problem. Use your second-degree polynomial to estimate У(1).

Consider a series solutions about the point x = 0 for the ODE: y" - xy + xy = 0 (i) Find a recurrence relationship for the coefficients of the series. (ii) Using (i) to evaluate the first four non-zero terms of each of the two linearly independent solutions and state the general solution for y(x).

The Legendre’s differential equation is  (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].