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Homework Assignment #3 - due Wednesday, 10/7, 21:00 UTC
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
Solve eqn. 2 on (p.13-1) for .
Problem Statement: Given a first integral of a L2_ODE_VC, solve for .
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:
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,
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
From (p.13-3), for the case (N1_ODE) . Show that . Hint: Use .
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:
Find in terms of .
Find in terms of
From (p.13-3), for the case (N2_ODE) show:
5.4) Relate eqn. 5 to eqs. 4&5 from p.10-2.
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 .
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.
From (p.15-2), plot the shape function .
Problem Statement: From (p.16-2), show that
Factor out and re-arrange terms in ordre of derivative,
Factor out and re-arrange terms in order of derivative.
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 Statement: From (p.17-4 ) obtain equation 2 from p.17-3
using the integrator factor method.
Problem Statement: From (p.18-1 ), develop reduction of order method using the following algebraic options
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
Joe Gaddone 16:46, 3 October 2009 (UTC)
Egm6321.f09.Team2.sungsik 21:22, 4 November 2009 (UTC)