MA2051 - Ordinary Differential Equations
Review For Test II 





1. 
Work with the following differential equation:








  (a)


  Find two linearly independent solutions for the
  corresponding homogeneous equation.
  Prove that your solutions really are linearly independent. 





  (b)

 
Find a particular solutionfor the original equation.




  (c)

 
  Find the solution for the equation which satisfies
  . 




  (d)


  Does the solution have a transient component?  
  If it does, estimate how long it takes for the
  magnitude of this component to become less than .




  (e)


  Does the equation have a limiting periodic solution?  If it does,
  identify the amplitude and period of this limiting solution.




  (f)


  Determine initial values ( and ) so that the
  solution to the initial value problem contains no transient
  component.






2.  Consider the two functions 








  (a)


  Find a linear, constant-coefficient, homogeneous
  differential equation that has  and  as solutions.


  (b)


  Determine initial conditions  and 
  so that  is the  solution for the differential
  equation that you found in (a).






3.
A mass of 6 kg is suspended from a spring that extended  m
when a 1 kg mass was attached.  Assume that friction damping
can be ignored (until part (d)).  You pull the mass 
down5 cm and release it from rest.






  (a)


  Write down the mathematical model for this
  system.  Use the given data to
  specify the numerical values for the parameters in the model. 


  (b)


  Determine the natural frequency,the 
  period,and the amplitude
  of the motion of the mass.


  (c)


  How much would you have to increase the mass in order to
  double the period?  How much would you have to 
  increase the mass in order to double the the amplitude?


  (d)


  Take the same spring-mass system and immerse it
  in a vat of molasses. It is no longer reasonable to 
  ignore the force of friction damping.
  For what values of the
  damping coefficient p will the spring still exhibit 
  oscillations about the steady state?   Explain your answer.






4. Consider the following system of first-order
differential equations:






  (a)

 
  Find two solution pairs and verify that they are linearly independent.
  Give the general solution for the system. 


  (b)

  
  Find the solution satisfying initial conditions
   and . 


  (c)


  Graph the solution obtained in part (b) in the 
  --plane, you would find that it is a spiral going 
  out away from the origin.


  (d)


  Using only the differential equation (and not the actual solution),
  show that the quantity  is actually increasing
  along any solution curve and so the solution must be heading
  away from the origin.






5. 
Use two steps of Euler's method to approximate the solution to


at t = 1.  Compare your numerical result with the exact solution
obtained in problem #1 above.  What is the error?







      © 1996 by Will Brother.
      All rights Reserved. File last modified on April 25, 1996.