Differential Equations     Math 285    Practice Hour Exam 3   Fall, 2011

 

 

 

 

 

 

4.

 

 

 

                                                                                          ∞

Consider the following differential equation  x’’ + k x = Σ (1/n2) cos (2n+1)πt/3

                                                                                        n=1

 

where  x(t) is a function of  t  on the entire line.

 

a.        For which  k  from the following list does pure resonance occur?

 

                    ( k =  1,  π2 ,   5π/3,  5/3,  49 π2/9)

 

b.      Find a solution of the equation for  k = 2.

                                                                ∞

Answers:  a. π2 , 49 π2/9      b.  x(t)  =  9 Σ 1/[ 18 – (2n + 1)2 π2] cos (2n + 1) πt/3

                                                               n=1

 

 

 

5.

 

Find the eigenvalues and eigenfunctions of the problem   y’’ + λ y = 0    y’(0) = 0,   y(3) = 0

 

Answers:  λn  =  (π2/36) (2n + 1)2   for   n ≥ 0,       yn(x) = cos (2n + 1) πx/6

 

 

 

 

6.

 

 

 

Solve the initial-boundary-value problem:  ut = 10 uxx,    0 < x < 2,   t > 0

Formula for the general solution is given on exam.

 

 ux(0,t) =  ux(2,t) = 0,   u(x,0) =  5 – 3 cos 2πx  +  4 cos 10πx

 

Answer:  u(x,t)  =  5 – 3 exp(-40π2t) cos(2πx) + 4 exp(-1000π2t) cos(10πx)

 

 

 

 

7.

 

Solve the initial-boundary-value problem:  utt = 25 uxx,    0 < x < 2,   t > 0

Formula for the general solution is given on exam.

 

 u(0,t) =  u(2,t) = 0,    u(x,0) =  5 sin 2πx,   ut(x,0)  =  -2 sin 10πx

 

Answer:  u(x,t)  =  5 cos10πt sin 2πx – (1/25π) sin 50πt sin 10πx