MATH201 Lecture : ch5.pdf
16 Apr 2014
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So far we have given a systematic procedure for constructing fundamental solutions only if the equation had constant coe cients. There is no known type of second order, linear di erential equation, apart from constant coe cient equations and those reducible to them by a change of variable, which can be solved in terms of elementary functions. We therefore resort to nding solutions in the form of power series. An expression of the form uk = u0 + u1 + + uk + , where {uk} the series as k=0 is a sequence of real numbers, is called an in nite series. Sn := uk = u0 + u1 + + un. Each in nite series has associated with it an in nite sequence of partial sums {sn} n=0. An in nite series p k=0 uk is said to converge if lim n . Sn = lim n uk = l, n.
