Page 326 - Schaum's Outline of Differential Equations

CHAPTER 32

Second-Order

Boundary-Value

Problems

STANDARD FORM
A boundary-value problem in standard form consists of the second-order linear differential equation

and the boundary conditions

where P(x), Q(x), and (f)(x) are continuous in [a, b\ and a x, a^, j\, J3 2, Ji, and / 2 are all real constants.
Furthermore, it is assumed that a x and /Jj are not both zero, and also that a^ and J3 2 are not both zero.
The boundary-value problem is said to be homogeneous if both the differential equation and the boundary
conditions are homogeneous (i.e. (f)(x) = 0 and Ji=Ji= 0). Otherwise the problem is non-homogeneous. Thus
a homogeneous boundary-value problem has the form

A somewhat more general homogeneous boundary-value problem than (32.3) is one where the coefficients P(x)
and Q(x) also depend on an arbitrary constant ^,. Such a problem has the form

Both (32.3) and (32.4) always admit the trivial solution y(x) = 0.

309
Copyright © 2006, 1994, 1973 by The McGraw-Hill Companies, Inc. Click here for terms of use.