1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 53

15.Solve foru(x).Notetheinterval.

d^4 u

dx^4

+ k

EI

u=w, 0 <x<∞ (wis constant),

u( 0 )= 0 ,

d^2 u

dx^2 (^0 )=^0.

16.Show that any two of the four functions sinh(λx),sinh(λ(a−x)),
cosh(λx),cosh(λ(a−x))are independent solutions of the differential
equation
φ′′−λ^2 φ= 0.

17.In this problem,uis the temperature in a wall composed of two sub-
stances. Findu(x).
d^2 u
dx^2 =^0 ,^0 <x<αa and αa<x<a,
u( 0 )=T 0 , u(a)=T 1 ,

κ 1 du

dx

(αa−)=κ 2 du

dx

(αa+),

u(αa−)=u(αa+).

The last two conditions say that the heat flow rate and the temperature

are both continuous across the interface atx=αa.

18.Find the general solution of the differential equation

1

x^2

d

dx

(

x^2 du

dx

)

+ku= 0

for the casesk=λ^2 andk=−p^2 .(Hint:Letu(x)=v(x)/xand find the

equation thatv(x)satisfies.)

19.Find the solution of the boundary value problem

ex

d

dx

(

ex

du

dx

)

=− 1 , 0 <x<a,

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

20.Solve the boundary value problem

1

r

d

dr

(

rdu

dr

)

=−rk, 0 <r<a,

u( 0 ) bounded and u(a)= 0.