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



Instructions: Do your work on the paper provided.  Put 
your name, section number and PLA's name on that paper.  Work 
neatly.  Show your work. JUSTIFY YOUR ANSWERS.





1 25 pts 

  A large vat is 
filled with a solution of a chemical whose concentration at time t is 
c(t) moles/cm tex2html_wrap_inline42 .  Each second a fixed fraction k of the chemical 
is lost through reactions with other species in the vat.  
Simultaneously, new chemical is being added at a rate of a 
moles/cm tex2html_wrap_inline42 -s.  Which of the following is a reasonable model for the 
concentration c(t)?  Justify rejecting or accepting each 
choice.


    
 c'(t) - kc(t) = a,  tex2html_wrap_inline54 


    
 c'(t) - kc(t) = -a,  tex2html_wrap_inline54 


    
 c'(t) + kc(t) = a,  tex2html_wrap_inline54 


    
 c'(t) + kc(t) = -a,  tex2html_wrap_inline54 





2 40 pts 

  Money in a bank 
account is earning interest at the rate of 100I% per day.  A 
regular withdrawal of W dollars is made each day.  A model for the 
balance m in the account is m' - Im = -W,  tex2html_wrap_inline76 .


    
A
 Solve the initial-value problem m' - Im = -W,  tex2html_wrap_inline76 .


    
B
 Find the steady state(s) of this equation.


    
C
Do you believe the steady state value(s) to be reasonable?  
    Explain.


    
C
 Determine the stability of the steady state(s).


    
D
    Do you believe the stability conclusion(s) to be reasonable?  
    Explain.





3 35 pts 

 Consider the 
differential equation y' + 6y = 18 t - 9.


    
A
 Find a general solution of y' + 6y = 18 t - 9.


    
B
 Solve the initial-value problem y' + 6y = 18 t - 9, y(0) = 
    4.


    
C
 Use two steps of Euler's method to estimate the solution 
    of y' + 6y = 18 t - 9, y(0) = 4 at t = 2.  (You don't need a 
    calculator.)




    
 (5 points extra credit) Write (but don't evaluate) an expression whose value is the 
    error in the Euler approximation you just found.


 




  

    

© 1996 by Will Brother.
All rights Reserved. File last modified on September 29, 1996.