MA2051 - Ordinary Differential Equations
Sample Exam 2 Solutions - A96

Second Exam - Originally Given 1993 A Term 











1






A




tex2html_wrap_inline70  dynes/cm; the model is 200 x''
    + 24,500 x = 0, x(0) = 0, x'(0) = -6.


B


Use characteristic equations:  tex2html_wrap_inline78 ;  tex2html_wrap_inline80 . Hence, a general solution is  tex2html_wrap_inline82 .  The initial conditions force  tex2html_wrap_inline84 ,
     C2 = -.06/11.07 = -.00542 .  Hence,  tex2html_wrap_inline88 .


C


To find the period, solve  tex2html_wrap_inline90  to find  tex2html_wrap_inline92  s.


D


The maximum displacement occurs in  tex2html_wrap_inline88  when the sine function is  tex2html_wrap_inline96 .  Hence, the maximum
    displacement (or amplitude of the motion) is 1.85 cm.










2




A


Use undetermined coefficients.  First guess  tex2html_wrap_inline98 .  Because these functions are solutions of the
    homogeneous equation, the correct guess is  tex2html_wrap_inline100 .  Substitute and collect coefficients of sine and
    cosine to find A = 0, B = 1/4.  Hence,  tex2html_wrap_inline106 .


B


Using the particular solution just found and the
    given homogeneous solutions,  tex2html_wrap_inline108 .


C


Because of the leading factor of t in  tex2html_wrap_inline112 , 
    the solution grows without bound.  The system is being forced at
    its resonant frequency.










3






A


r = -4, 2


B


tex2html_wrap_inline116 .


C


Choose  tex2html_wrap_inline118 ,  tex2html_wrap_inline120 .










4






A


Write  tex2html_wrap_inline122 .




B


tex2html_wrap_inline124 


C


Let  tex2html_wrap_inline126 .  Then  tex2html_wrap_inline128 
    and
    
 displaymath62 


  •   tex2html_wrap_inline130 .  Using partial fractions, we find
    
     displaymath63 
    
    Then  tex2html_wrap_inline132 ,  tex2html_wrap_inline134 , and
    
     displaymath64 
    
    Hence, the solution is
    
     displaymath65 
    

    • 

    




    
  
    
    
    
© 1996 by     Will Brother.
All rights Reserved. File last modified on December 6, 1996.