MA2051 - Ordinary Differential Equations
Solutions to Sample Exam #2 - A96

Originally Given 1995 D Term 












	1






	A,B,C)
y' = -ky + Q; -k y is decay (decreasing y because of 
negative sign) proportional to mass, and Q is constant rate of 
increase (assuming Q > 0).
tex2html_wrap_inline51  constant forces  tex2html_wrap_inline53  and  tex2html_wrap_inline55 .


	D)


Use  tex2html_wrap_inline57  as a particular solution or apply undetermined 
coefficients to this linear, constant coefficient equation to 
obtain  tex2html_wrap_inline59 .


	E)


 Solve y' + ky = 0 for  tex2html_wrap_inline63  via characteristic equations to 
    obtain  tex2html_wrap_inline65 .  Use  tex2html_wrap_inline67  from above to write  tex2html_wrap_inline69 .


	F)

 Use  tex2html_wrap_inline71  in  tex2html_wrap_inline73  to find C:  tex2html_wrap_inline77  yields
     tex2html_wrap_inline79 .


	G)


 The steady state  tex2html_wrap_inline55  is stable; see the graph 
    above or use the solution in the previous part to argue  tex2html_wrap_inline83  for all  tex2html_wrap_inline85 .










	2




  •  Apply Euler with unknown step size ,  tex2html_wrap_inline89 :
 tex2html_wrap_inline91 .
Hence, the population of the U.S. will double from 205 million to 410 
million in about the year 1970 + 100 = 2070.

  •  (Same as 3.8/6 with y replacing u, t replacing x.)  Apply
undetermined coefficients for a polynomial (4t) and an exponential 
( tex2html_wrap_inline103 ) forcing term; first guess is  tex2html_wrap_inline105 .  
But a solution of the homogeneous equation y' - 4y = 0 is  tex2html_wrap_inline103 .  
Hence, the final form should be

     displaymath39 

    • 

    

    

    
	3

    
Substitution yields  tex2html_wrap_inline111 .

    


    
  
    
    
    
© 1996 by     Will Brother.
All rights Reserved. File last modified on September 16, 1996.