1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

56 Chapter 0 Ordinary Differential Equations

whereDis thediffusion constant; and (2) when the sulphur dioxide reacts

with water, it “disappears” at a rate proportional to its concentration, say

ku(x)(in units of mass per unit time per unit volume).

29.The sulphur dioxide concentration in the air in a deep layer of snow

satisfies this boundary value problem in equilibrium conditions:

d^2 u

dx^2 −a

(^2) u= 0 , 0 <x,
u( 0 )=C 0.
Here,C 0 is the concentration in freely circulating air. Add an appropriate
boundedness condition and solve foru(x).
30.In “Mechanical properties of thin films from the load deflection of long
clamped plates” [V. Ziebart et al.,J. of Microelectromechanical Systems, 7
(1998): 320–327] this boundary value problem is studied:
d^4 w
dx^4 −γ
2 d^2 w
dx^2 =P, −

1

2 <x<

1

2 ,

w

(

±^1

2

)

= 0 , dw

dx

(

±^1

2

)

= 0.

The variables arewdeflection,xdistance measured across the short di-

mension; and the parameters arePpressure beneath the plate andγ^2

effective stress, all dimensionless. Find the general solution of the differ-

ential equation.

31.(Continuation) Solve the boundary value problem in Exercise 30.

32.(Continuation) The parameterγ^2 is related to stress, which is related to

deflection. It must satisfy the equation

γ^2 =S 0 +

∫ 1 / 2

0

(dw

dx

) 2

dx.

Use your solution to find a single explicit equation thatγsatisfies.

33.(Continuation) IfS 0 is negative in the equation of Exercise 32,γ^2 might

be negative, say,γ^2 =−λ^2. If this is the case, is there a value ofλfor

which the solution breaks down?

34.Suppose thatu(t)is a function, not identically 0, for which

u′′

u =constant>^0.