38 Chapter 0 Ordinary Differential Equations
(Here,Eis Young’s modulus andIis the second moment of the cross sec-
tion.) Solve this problem ifw(x)=w 0 , constant.
19.If the beam of Exercise 18 is built into a wall at the left end and is unsup-
ported at the right end, the boundary conditions become
u( 0 )= 0 , u′( 0 )= 0 , u′′(a)= 0 , u′′′(a)= 0.
Solve the same differential equation subject to these conditions.0.4 Singular Boundary Value Problems
A boundary value problem can be singular in two different ways. In one case,
an endpoint of the interval of interest is a singular point of the differential
equation. In the other, the interval is infinitely long.
Regular Singular Point
Recall that a pointx 0 is a (regular) singular point of the differential equation
u′′+k(x)u′+p(x)u=f(x)if the products
(x−x 0 )k(x), (x−x 0 )^2 p(x)both have Taylor series expansions centered atx 0 but eitherk(x)orp(x)or
both become infinite asx→x 0. For example, the pointx 0 =1 is a regular
singular point of the differential equation
( 1 −x)u′′+u′+xu= 0.In standard form, the equation is
u′′+ 1 −^1 xu′+ 1 −xxu= 0.Since both
k(x)=1
1 −x and p(x)=x
1 −x
become infinite atx=1, but(x− 1 )k(x)and(x− 1 )^2 p(x)both have Taylor
series expansions about the centerx=1, the pointx 0 =1 is a regular singular
point. Another convenient example is provided by the Cauchy–Euler equation
of Section 1, which has a regular singular point at the origin.
This situation typically arises when a boundary point is a mathematical
boundary without being a physical boundary. For instance, a circular disk of