Series solutions of differential equations

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form 

The Frobenius or series solution method for solving equation (G.l) assumes that the solution may be expressed as a power series in x 

A second-order differential equation has two solutions of the form of equation (G.2), each with a different set of values for the constant s and the coefficients ak. 

Not all differential equations of the general form (G.l) possess solutions which can be expressed as a power series (equation (G.2)).1 However, the differential equations encountered in quantum mechanics can be treated in this manner. Moreover, the power 

1 For a thorough treatment see F. B. Ftildebrand (1949) Advanced Calculus for Engineers (Prentice-FIall, New York). 

The terms are arranged in order of ascending powers of x to obtain 

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In developing series solutions of differential equations and in other formal calculations it is often convenient to make use of properties of gamma and beta functions. The integral... 

Section 4.1 Power-Series Solution of Differential Equations 63... 

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