Problem Description:

Solve the following variable coefficient linear system:

Problem Solution:

In standard form, the original linear ODE becomes

This form is similar to that observed in Example 7.2, thus we can use the extended power series method, with

and the derivative relations

Substitution of these expressions into the base ODE gives

Collecting like terms gives

and, with a little algebra, this becomes

Now, working on the second term, we let m = p+2 or p = m-2. Then the second term becomes

Now to get the desired recurrence relation between a_m+2 and a_m, we have

and

Now for r = r₁ = 1, we have

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

In fact, for all odd m, a_m = 0 since a₁ = 0, and for all even m > 0, a_m = 0 since a₂ = 0. Therefore, a₀ is the only non-zero term. This gives the simple result that

Performing similar manipulations for r = r₂ = -1, we have

but for m = 0, this term becomes undefined (i.e.). This is clearly not allowed. Therefore, r = -1 is not a valid root. Actually, this behavior was not completely unexpected since r₁ and r₂ differ by an integer. In this case, the phenomenon observed here is not uncommon.

Substitution of these expressions into the defining ODE gives

Expanding and performing the indicated algebraic operations give

or

This equation is separable and it can be simplified considerably using a partial fraction expansion technique, as follows:

Separating variables gives

where the A, B, and C constants are determined from

Therefore, we have the simplified form

This expression can be integrated directly, giving

or

One final integration gives

Therefore, the second linearly independent solution becomes

Finally, the linear combination of the two independent solutions gives the desired general solution for this problem, or