1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

0.3 Boundary Value Problems 35

b. d

(^2) u
dx^2
+λ^2 u=0, du
dx
( 0 )=0, u(a)=0;
c.
d^2 u
dx^2 +λ
(^2) u=0, du
dx(^0 )=0,
du
dx(a)=0.
4.Verify, by differentiating and substituting, that
u(x)=c′+^1
μ
cosh

(

μ(x+c)

)

is the general solution of the differential equation (5). (Hereμ=w/T.

The graph ofu(x)is called acatenary.)

5.Find the values ofcandc′for which the functionu(x)in Exercise 4 satisfies
the conditions
u( 0 )=h, u(a)=h.

6.A beam that is simply supported at its ends carries a distributed lateral
load of uniform intensityw(force/length) and an axial tension loadT
(force). The displacementu(x)of its centerline (positive down) satisfies
the boundary value problem here. Findu(x).
d^2 u
dx^2 −

T

EIu=−

w

EI

Lx−x^2

2 ,^0 <x<L,

u( 0 )= 0 , u(L)= 0.

7.The temperatureu(x)in a cooling fin satisfies the differential equation

d^2 u

dx^2

=hC

κA

(u−T), 0 <x<a,

and boundary conditions

u( 0 )=T 0 , −κdudx(a)=h

(

u(a)−T

)

.

That is, the temperature at the left end is held atT 0 >Twhile the surface

of the rod and its right end exchange heat with a surrounding medium at

temperatureT.Findu(x).

8.Calculate the limit asatends to infinity ofu(x), the solution of the prob-
lem in Exercise 7. Is the result physically reasonable?

9.In an electrical heating element, the temperatureu(x)satisfies the bound-
ary value problem that follows. Findu(x).
d^2 u
dx^2 =

hC

κA(u−T)−

I^2 R

κA,^0 <x<a,

u( 0 )=T, u(a)=T.