Problem Description:

Solve the following linear variable coefficient system:

Problem Solution:

The 2^nd order ODE in standard form is

In this case, the full coefficients are not analytic at x = 0, but b(x) and c(x) are analytic in the general form,

Therefore, we need the extended power series representation for this problem, with

and the corresponding derivative relationships

Upon substitution, the original ODE becomes

Combining all the coefficients for the terms with x^m+r, we let m = p+1 and p = m-1, giving

Now removing the p = -1 term from the sum gives

Requiring that gives the indicial equation,

or

The desired recurrence relationship between a_m+1 and a_m can be obtained by setting each coefficient of x^m+r+1 to zero (because we have a homogeneous system), or

and for r₁ = 1/4, this relationship gives

Therefore, for r₁ = 1/4, we have,

Letting m = 0, 1, etc., gives

Therefore, since a₀ is arbitrary we can write the final form for y₁(x) as

A second linearly independent solution is obtained in a similar manner by setting r = r₂ = -1/2. For this case, after substitution of r₂ into the above balance relationships and some careful algebraic manipulation, we have

Therefore, for r = -1/2, we have

Letting m = 0, 1, etc., gives

Again, since a₀ is arbitrary, y₂(x) becomes

and the final solution is simply a linear combination of the two linearly independent solutions, or

which, when written explicitly, gives