Differential Equations   Math 285  Hour Exam 2   Fall, 2011

 

 

 

 

 

 

 

 

 

 

 

4.

 

 

 

A mass  10 g  stretches a vertically suspended spring  140 cm.  The mass moves in a

medium that imparts a resistance force of 40 dynes when the speed is 1 cm/sec.  The

mass is pulled up 4 cm and set in motion with initial downward velocity of 20 cm/sec.

(All the quantities are given in the metric cm-gram-sec system of units.   “Dyne” is the

unit of force in this system.  Recall the acceleration due to gravity is  g = 980 cm/sec2.

Since calculators are not to be used, the answers may include  π,  √,  etc.)

 

a.        Set up an initial value problem for the position of the mass relative to its

 static equilibrium after time  t.  Use the downward direction of the coordinate

 axis.

 

b.      Solve the initial value problem in part a.

c.       Find the circular (pseudo) frequency, time varying amplitude, and phase

of the motion.

 

Answers:  a.  10 x” + 40 x’ + 70 x  =  0    x(0) = -4,   x ‘ (0)  =  20

 

b.  x(t) = ( - 4 cos √3 t  +  4 √3 sin √3 t ) e -2t

 

c.  ω = √3     Amp(t)  =  8 e -2t     α  =  2π/3

 

 

 

 

 

 

5.

 

Find a general solution to the equation

 

       x3 y”  - 3 x2 y’  + 3 x y  =  2 x 3

 

given that the functions   x  and  x3  satisfy the corresponding homogeneous equation.

(Use variation of parameters.  Pay attention to the coefficient of y”.)

 

Answer:  y  =  -2 x2  +  C1 x  +  C2 x 3