50 Chapter 0 Ordinary Differential Equations
- d
(^2) u
dx^2
−γ^2 u=f(x), −∞<x<∞,
u(x)bounded asx→±∞.
- Use the Green’s function of Exercise 5 to solve the problem
1
ρ^2
d
dρ(
ρ^2 du
dρ)
= 1 , 0 ≤ρ<c,u(c)= 0 ,
and compare with the solution found by integrating the equation directly.
10.Use the Green’s function of Exercise 8 to solve the problem
d^2 u
dx^2−γ^2 u=−γ^2 , −∞<x<∞,
u(x) bounded asx→±∞,
and compare with the result found directly.
11.Use the Green’s function of Exercise 1 to solve the problem stated there, iff(x)={ 0 , 0 <x<a/2,1 , a/ 2 <x<a.12.In confirmation of the theorem, show that the homogeneous problema
has a nontrivial solution; problembhasnosolution(existencefails);and
problemchas infinitely many solutions (uniqueness fails).
a.u′′+u=0, u( 0 )=0, u(π )=0,
b.u′′+u=−1, u( 0 )=0, u(π )=0,
c. u′′+u=π− 2 x, u( 0 )=0, u(π )=0.
13.Consideringzto be a parameter(l<z<r), define the functionv(x)=
G(x,z)withGasinEq.(17).Showthatvhas these four properties, which
aresometimesusedtodefinetheGreen’sfunction.
(i) vsatisfies the boundary conditions, Eqs. (2) and (3), atx=landr.
(ii)vis continuous,l<x<r.(Thepointx=zneeds to be checked.)
(iii)v′is discontinuous atx=z,andhlim→ 0 +(
v′(z+h)−v′(z−h))
= 1.
(iv)vsatisfies the differential equationv′′+k(x)v′+p(x)v=0 forl<
x<zandz<x<r.