University of Florida/Egm6321/f09.Team2/HW3

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Homework Assignment #3 - due Wednesday, 10/7, 21:00 UTC

Problem 1Edit

Find   such that eqn. 1 on (p.13-1) is exact. A first integral is   where   are constants.


Problem Statement: Given a L2_ODE_VC  

Find (m,n) from the integrating factor (xm,yn) that makes the equation exact.

A first integral is  


 
 
 

 

 

 

 

 
 

 

 

 

 

Problem 2Edit

Solve eqn. 2 on (p.13-1) for  .


Problem Statement: Given a first integral  of a L2_ODE_VC, solve for  .

  (1)

where k1 and k2 are const, and  


Eq. (1) is in the form   where

 

 

so it satisfies the 1st condition of exactness.

Check if   for the 2ndcondition of exactness

 

 

  so we do not satisfy the 2nd condition of exactness.

We must apply the integrating factor method for a L1_ODE_VC.

 , divide by x to obtain the form:

  where:

 

 

From our solution of a general non-homogeneous L1_ODE_VC p.8-1

 

 

From p.8-2 Eq. (4)

 

Use the product rule of integration  

 

In our example   so,

 

 

Problem 3Edit

From (p.13-1), find the mathematical structure of   that yields the above class of ODE.


 

 

 

 

 

 

Take the integral of  

 

Substitute back into the equation for  

 

Rearrange the terms to obtain

  where,

 
 
 

 

Problem 4Edit

From (p.13-3), for the case   (N1_ODE)  . Show that  . Hint: Use  .
Specifically:
4.1) Find   in terms of  
4.2) Find   in terms of  ( )
4.3) Show that  .


Problem Statement: Given a N1_ODE, for the case n=1  

Show that   Hint: 


 

 

 

4.1Edit

Find   in terms of  .

 

 

4.2Edit

Find   in terms of  

 

 

4.3Edit

Show that  

 

 

Problem 5Edit

From (p.13-3), for the case   (N2_ODE) show:
5.1) Show  
5.2) Show  
5.3)  
5.4) Relate eqn. 5 to eqs. 4&5 from p.10-2.

Problem 6Edit

From (p.14-2), for the Legendre differential equation  ,
6.1 Verify exactness of this equation using two methods:
6.1a.) (p.10-3), Equations 4&5.
6.1b.) (p.14-1), Equation 5.
6.2 If it is not exact, see whether it can be made exact using the integrating factor with  .

Problem 7Edit

From (p.14-3), Show that equations 1 and 2, namely
7.1   functions of  ,  . and
7.2   functions of  .
are equivalent to equation 3 on p.3-3.

Problem 8Edit

From (p.15-2), plot the shape function  .

Media:Graph1.pdf

Problem 9Edit

Problem Statement: From (p.16-2), show that
 
 


 

Replace  

  'Chain Rule'

 

 

 

 

Factor out   and re-arrange terms in ordre of derivative,

 


 

 

Replace  

 

 

 

 

 

 

 

Factor out   and re-arrange terms in order of derivative.

 

Problem 10Edit

Problem Statement: From (p.16-4 ) Solve equation 1 on p.16-1,   using the method of trial solution   directly for the boundary conditions  
Compare the solution with equation 10 on p.16-3. Use matlab to plot the solutions.


Problem 11Edit

Problem Statement: From (p.17-4 ) obtain equation 2 from p.17-3   using the integrator factor method.


Problem 12Edit

Problem Statement: From (p.18-1 ), develop reduction of order method using the following algebraic options

 

 

 


Problem 13Edit

Problem Statement: From (p.18-1 ), Find   and   of equation 1 on p.18-1 using 2 trial solutions:

 

 

Compare the two solutions using boundary conditions   and   and compare to the solution by reduction of order method 2. Plot the solutions in Matlab.


Contributing Team MembersEdit

Joe Gaddone 16:46, 3 October 2009 (UTC)

Matthew Walker

Egm6321.f09.Team2.sungsik 21:22, 4 November 2009 (UTC)