Let y = > anx" be the power series solution in x to the initial value Σ n=0 5 problem 2y" + 5xy' + (4+ 2x?)y = 0 with y(0) = 3 and y'(0) = -2. Find ) Anr". n=0 the 5th degree polynomial approximation to the solution.

Assume the solution to the differential equation that follows is the power series y = Σ an (x + 3)". y"+xy'+y= 0, y(-3) = 3, y'(-3) = 2. The first few terms of the series solution are: y = ao + a1(x + 3) + a2(x+3)² + α3(x+3) ³ + α4(x+3)+ where ао a1 a2 a3 a4 = = = = = n=0

3. [ solution around xo ] Find the first three non - zero terms of each linearly independent power series = 0 for (x +2)y" +3xy' - y=0

c) Consider the differential equation 1 y" + y +y = 0. Assuming that the solution can be written as a power series, y = ao + a1x + a2x² + · · · + anx" + . ..= anx" n=0 show that a1 = 0 and that the other constants in this series must satisfy 1 an-2. an Hence derive the power series for y(x) up to and including powers of x6 assuming the constant ao = 1.