1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

54 Chapter 0 Ordinary Differential Equations

21.Solve the differential equation

d^2 u

dx^2

=p^2 u, 0 <x<a,

subject to the following sets of boundary conditions.

a. u( 0 )=0, u(a)=1;

b. u( 0 )=1, u(a)=0;

c. u′( 0 )=0, u(a)=1;

d. u( 0 )=1, u′(a)=0;

e. u′( 0 )=1, u′(a)=0;

f.u′( 0 )=0, u′(a)=1.

22.Solve the integro-differential boundary value problem

d^2 u

dx^2

=γ^2

(

u−

∫ 1

0

u(x)dx

)

, 0 <x< 1 ,

du

dx

( 0 )= 0 , u( 1 )=T.

Hint: Look for a solution in the form

u(x)=Acosh(γx)+Bsinh(γx)+C.

23.Use a variation of parameters to find a second independent solution of

the following differential equation. One solution is given in parentheses.

d^2 u

dx^2 −

2 x

1 −x^2

du

dx+

2

1 −x^2 u=^0 (u=x).

24.By applying the method of variation of parameters, derive this formula

for a particular solution of the differential equation

d^2 u

dx^2

−γ^2 u=f(x),

u(x)=

∫x

0

f(x′)sinhγ(x−x

′)

γ dx

′.

25.The absolute temperatureu(x)in a cooling fin that radiates heat to a

medium at absolute temperatureTobeys the differential equationu′′=

γ^2 (u^4 −T^4 ). Solve the special version in the boundary value problem