34 Chapter 0 Ordinary Differential Equations
multiple ofπ,sincesin(π )=0, sin( 2 π)=0, etc., and integer multiples ofπ
are the only arguments for which the sine function is 0. The equationλa=π,
in terms of the original parameters, is
√
P
EIa=π.
It is reasonable to think ofE,I,andaas given quantities; thus it is the force
P=EI(π
a) 2
,
called the critical or Euler load, that causes the buckling. The higher critical
loads, corresponding toλa= 2 π,λa= 3 π, etc., are so unstable as to be of no
physical interest in this problem.
The buckling example is one instance of aneigenvalueproblem. The gen-
eral setting is a homogeneous differential equation containing a parameterλ
and accompanied by homogeneous boundary conditions. Because both dif-
ferential equations and boundary conditions are homogeneous, the constant
function 0 is always a solution. The question to be answered is: What values
of the parameterλallow the existence of nonzero solutions? Eigenvalue prob-
lems often are employed to find the dividing line between stable and unstable
behavior. We will see them frequently in later chapters.
EXERCISES
- Of these three boundary value problems, one has no solution, one has
exactly one solution, and one has an infinite number of solutions. Which
is which?
a.d^2 u
dx^2 +u=0, u(^0 )=0, u(π )=0;b.d^2 u
dx^2 +u=1, u(^0 )=0, u(^1 )=0;c. d(^2) u
dx^2 +u=0, u(^0 )=0, u(π )=1.
- Find the Euler buckling load of a steel column with a 2 in.×3in.rectan-
gular cross section. The parameters areE= 30 × 106 lb/in.^2 ,I=2in.^4 ,
a=10 ft. - Find all values of the parameterλfor which these homogeneous boundary
value problems have a solution other thanu(x)≡0.
a. d(^2) u
dx^2 +λ
(^2) u=0, u( 0 )=0, du
dx(a)=0;