University of Florida/Egm6321/f09.Team2/HW3

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

Problem 1[edit]

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 2[edit]

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 3[edit]

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 4[edit]

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:


4.1[edit]

Find in terms of .

4.2[edit]

Find in terms of

4.3[edit]

Show that

Problem 5[edit]

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 6[edit]

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 7[edit]

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 8[edit]

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

Media:Graph1.pdf

Problem 9[edit]

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 10[edit]

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 11[edit]

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


Problem 12[edit]

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





Problem 13[edit]

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 Members[edit]

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

Matthew Walker

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