The first three non – zero terms in a power series expansion of the solution for IVP y' – xy' – y = 0 with y(0) = 0, y (0) = -2 are given by : - 21 8 ,4

Assume y(x) = ax" for #2, 3, 4, 5 n=0 5. Find at least the first four nonzero terms in a power series expansion of the I.V.P. y'-e'y=0; y(0)=1 1

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).