MA2051 - Ordinary Differential Equations
Test 1 - 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 55 pts 

  A tall water tank 
has a small leak in its bottom.  The higher the water level in the 
tank, the faster the water leaks.  To be precise, a fixed fraction r 
of the volume V of water in the tank leaks from it each minute.  A 
hose is filling the tank at a rate of H l/min.


    
A
 Write an expression for the rate at which water leaks 
    from the tank.


    
B
 Write an expression for the rate at which water is added 
    to the tank.


    
C
 State a law of conservation of water.  Use it to derive the 
    governing differential equation dV/dt = -rV + H.


    
D
 Find the steady state(s) of this equation.  Explain the 
    physical significance of each steady state.


    
E
 Determine the stability of each steady state.







2 25 pts 

 Find the solution 
of  tex2html_wrap_inline51 , u(0) = 3.





3 20 pts 

 Which of the 
four curves labeled A-D could not possibly be a graph of the 
solution of  tex2html_wrap_inline55 , y(0) = 1?  Justify rejecting each 
unacceptable choice.



 



  

    

© 1996 by Will Brother.
All rights Reserved. File last modified on September 18, 1996.