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E1.4 A two-point boundary-value problem
Our final introductory example is provided by

with and given. This equation contains the parameter in two places:
multiplying the higher derivative, which is critical here (as we will see), and adjusting
the coefficient of the other derivative by a small amount. This latter appearance of the
parameter is altogether unimportant—the coefficient is certainly close to unity—and
serves only to make more transparent the calculations that we present.
Once again, we will start by seeking a solution which can be represented by the
series

so that we obtain

the shorthand notation for derivatives is again being employed. Thus we have the set
of differential equations

with boundary conditions written as

where and are given (but we will assume that they are not functions of The
general solution for is

but it is not at all clear how we can determine A. The difficulty that we have in this
example is that we must apply two boundary conditions, which is patently impossible
(unless some special requirement is satisfied). So, if we use we obtain

if, by extreme good fortune, we have then we also satisfy the second
boundary condition (on x = 1). Of course, in general, this will not be the case; let us
proceed with the problem for which Thus the solution using does