Solve the initial value problems in Problems 18 through 22. First make a substitution of the form t = x - a, then find a solution Σcntⁿ of the transformed differential equation. State the interval of values of x for which Theorem 1 of this section guarantees convergence.

(x² + 6x)y" + (3x + 9)y' - 3y = 0; y(-3) = 0, y'(-3) = 2

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THEOREM 1 Solutions Near an Ordinary Point Suppose that a is an ordinary point of the equation A(x)y" + B(x)y' + C(x) y = 0; (1) that is, the functions P = B/A and Q = C/A are analytic at x = a. Then Eq. (1) has two linearly independent solutions, each of the form y(x) = [cn (x-a)". n=0 (5) The radius of convergence of any such series solution is at least as large as the distance from a to the nearest (real or complex) singular point of Eq. (1). The coefficients in the series in (5) can be determined by its substitution in Eq. (1).