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SynopsisWe investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.

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Bordetella bronchiseptica was isolated from 12 cases of purulent bronchopneumonia from a captive koala colony near Brisbane, Queensland. An initial epizootic in March 1967 causing 13% mortality was followed by annual outbreaks causing 2% mortality usually in newly weaned and aged koalas in late winter to early spring. High population density and a low plane of nutrition in winter were thought to predispose to the occurrence of the disease.

ABSTRACT. We develop a hyperasymptotic expansion for solutions of the general second-order homogeneous differential equation with a singularity of arbitrary integer Poincare rank at infinity. This expansion is in terms of certain integrals which are generalisations of the hyperterminant integrals developed in other papers for the rank one case.