Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0.
a) Find the recurrence relations satisfied by the coefficients an of the power series solution.
b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the first two terms a0, a1.
c) Deduce the general form of the coefficients an and write down the general solution as a linear combination of 2 linearly independent power series solutions.
d) What is the radius of convergence of each one of the two linearly independent power series solutions.
f) Find the power series solution of the initial value problem y'' + xy' + 2y = 0. y(0) = 3, y' (0) = −1. g) Give the polynomial approximations of degrees 4 and 5 for the solution of the initial value problem in (f).
The Attempt at a Solution
For a) I let y=∑anxn
and through taking the derivative and substituting I got
∑xn[(n+2)(n+1)an+ 2 + nan +2an ]= 0
Which showed that the recurrence relation is
which simplifies to an+2 = -an/(n+1)
b) I found that
a2 = -a0
a4 = -a0/(3)
a6 = -a0/(3*5)
a8 = -a0/(3*5*7)
a3 = -a1/2
a5 = -a1/(2*4)
a7 = -a1/(2*4*6)
c) I know I have to try to find a solution where I can write a0 ∑ + a1 ∑
but I don't know any series that has the pattern that I observed so I am lost.