Problem Description:

Solve the following linear variable coefficient second order homogeneous system:

Problem Solution:

The original equation written in standard form is

Since the coefficients are analytic at x = 0, we let

Then, substitution into the original ODE gives

Now, we can work on the first term to shift the exponent to the highest power (i.e. x^m), and also force the summation to begin at m = 0. Letting p = m-2 or m = p+2, we have

where a₀ and a₁ are arbitrary coefficients because the coefficients of the x^-2 and x^-1 terms are already zero.

Substituting this result into the full balance equation gives

or

and

Therefore, letting m = 0, 1, 2, etc. gives

Therefore, the final solution, y(x), can be written as

where we have grouped all the terms that multiply the a₀ and a₁ coefficients separately. Here one recognizes two individual linearly independent solutions to the original ODE, or

where c₁ = a₀ and c₂ = a₁, and

Thus, the solution procedure is complete and the final result is represented as a linear combination of two linearly independent solutions, as expected.