MA2051 - Ordinary Differential Equations
Test 1 - Make-Up 2 Solutions - A96 









1


Concentration c is decreased by kc term, increased by a.


 Reject:  tex2html_wrap_inline54  means kc causes increase, not 
    decrease, in c.


 Reject: same reason as (a) or (d).


 Accept:  tex2html_wrap_inline60  means kc causes decrease in 
    c;  tex2html_wrap_inline66  means a causes increase in c.


Reject:  tex2html_wrap_inline72  means a causes decrease in 
    c.










2



Characteristic equation yields  tex2html_wrap_inline78 .  Undetermined 
    coefficients with  tex2html_wrap_inline80  constant yields  tex2html_wrap_inline82 .  Hence, 
     tex2html_wrap_inline84 .  IC forces  tex2html_wrap_inline86  or  tex2html_wrap_inline88 .  Solution of IVP is  tex2html_wrap_inline90 .

tex2html_wrap_inline92  constant yields  tex2html_wrap_inline94  or 
     tex2html_wrap_inline96 .


        This steady state is reasonable.  It represents a balance 
    between withdrawals at rate W and interest earned at rate  tex2html_wrap_inline100 .

The steady state  tex2html_wrap_inline96  is unstable.  From the 
    form of the general solution, every solution of this IVP can be 
    written  tex2html_wrap_inline104  for some C that depends on 
    the IC.  Since  tex2html_wrap_inline108  for large t, every 
    solution blows up (except the one with  tex2html_wrap_inline112 ).


        Alternate argument: Consider solutions of the form  tex2html_wrap_inline114 .  Substitute into the DE, use  tex2html_wrap_inline96 , and find p' = Ip.  Hence,  tex2html_wrap_inline120  for 
    all  tex2html_wrap_inline122 , and  tex2html_wrap_inline124  is unstable.


        Instability is reasonable.  If the balance is slightly above 
     tex2html_wrap_inline124 , then interest will outstrip withdrawals and m will 
    grow; vice-versa if the balance is slightly below  tex2html_wrap_inline124 .  
    (This problem has the same structure as the emigration model.)










3





  •  Characteristic equations yields  tex2html_wrap_inline132 .  Undetermined 
    coefficients with  tex2html_wrap_inline134  yields A = 3, B = -2, and  tex2html_wrap_inline140 .  Hence,  tex2html_wrap_inline142 .

  •  Use  tex2html_wrap_inline144  just found and the IC:  tex2html_wrap_inline146  forces 
    C = 6.  The solution of the IVP is  tex2html_wrap_inline150 .

  •  Use y' = -6y + 18t - 9,  tex2html_wrap_inline154 .

    
        y_1 &= &4 + [ -6 4 + 18 0 - 9 ] 1 
            = -29 
     
        y_2 &= &-29 + [ -6 -29 + 18 1 - 9 ] 1 
            = 154

    

    • 

    

    

    
Extra Credit

    

  •  error  tex2html_wrap_inline156  (or vice-versa)  tex2html_wrap_inline158 .  The exact solution y(2) is from part (b), the Euler
    approximation  tex2html_wrap_inline162  from part (c).

    


    
  
    
    
    
© 1996 by     Will Brother.
All rights Reserved. File last modified on September 27, 1996.