By Carl M. Bender

ISBN-10: 007004452X

ISBN-13: 9780070044524

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F or additional properties of the r function see Prob. 6. Example 3 General first-order linear homogeneous difference equations. +, = p(nJa. 5) j=l In this [ormula at is an arbitrary constant. 24): In a. +-, = In pin) + In a•. Letting b. In a. gives b•• I - b. = In pin). The left side of this equation is an exact discrete derivative. Therefore. the solution b. is obtained by computing the discrete integral: b. '. 1 A plot of r(x) for - 4 ~ x ~ 4. v(x) = y(O) exp U~ p(t) dt]. Example 4 General first-order linear inhomogeneous difference equations.

4) Example 4 Conversion of a scale-invariant equation to an equidimensional-in-x equation. (a) The Thomas-Fermi equation in part (a) of the previous example may be made equidimensional by substituting y = x- 'u. The resulting equation x'u' 6xu' + 12u u31 ' is indeed equidimensional. so the substitution x = e' makes it autonomous: unit) - 7u'(t) + 12u = U'I'. This equation is equivalent to the 'first-order equation ww'(u) 7w + 12u = U ll ', where w(u) = u'(t). Unfortunately, this first-order equation is too difficult to solve in closed form; approximate rather than exact analytical methods are appropria1e for understanding the Thomas-Fermi equation.

Thus, the general solutioll for a, is a. = (e, + nc,)(-Ir + C" Linear Independence and Wronskians Since a complete solution to a homogeneous linear difference equation is a linear combination of N linearly illdependent solutions, it is important to be able to test the linear independence of a set of N functions of n. , ... to be linearly independent for II in the interval n l ::; II ::; liz if the only solution to kl an + kl b. + ... = 0 (111 ::; n ::; liz) is k t k z ... = O. ) In the study of functions of a continuous variable x the Wronskian provides a simple way to examine the linear independence of a set of functions, There is also a Wronskian W.

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