User:Egm6321.f10.team4.Yoon/Mtg32

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_02_14_58_42.djvu | Mtg 32:]] Thu, 16 Sep 10

= Summary =

[[media: 2010_11_02_14_58_42.djvu | Page 32-1]]

 * IFM solves all L1_ODE_VC
 * Not so for L2-ODE-VC

If 1 Homogeneous Solution => Solve using variation of parameters
 * L2-ODE-VC

= Polynomial series as trial solution: Frobenius method =

[[media: 2010_10_28_14_54_20.djvu | Eqn(2) p.31-1 ]]=> use more general trial solution with c and r constants to be determined

Use Wolfram Alpha to find (c,r) $$\displaystyle _{\Rightarrow} $$ c=-1, r = $$\displaystyle _{\pm} $$ i; See [[media: 2010_10_28_14_54_20.djvu | p.31-2]].

[[media: 2010_11_02_14_58_42.djvu | Page 32-2]]
[[media: 2010_10_28_14_00_19.djvu | Eqn(2) p.30-3 ]] => Wolfram Alpha => one Valid solution

same as [[media: 2010_10_28_14_00_19.djvu | Eqn(2) p.30-3 ]] Note: [[media: 2010_11_02_14_58_42.djvu | Eqn (1)p.32-1]] is similar to starting pt for method of Frobenius(see K.2003)

L2-ODE-VC Trial solution: to be determined => 1 homogeneous solution as a series with factor d0(1 integer constant)

e.g., Legendre polynomials (finite series) as first homogeneous solutions of Legendre equation.

1st homogeneous solutions = $$\displaystyle [P_0(x), P_1(x), P_2(x), ...] $$ 2nd homogeneous solutions = $$\displaystyle [Q_0(x), Q_1(x), Q_2(x), ...] $$ Given a 1st homogeneous solution $$\displaystyle P_i(x), i=0,1,2,...,$$ corresponding 2nd homogeneous solution $$\displaystyle Q_i(x) $$ is obtained by variation parameters.

= Legendre functions =

[[media: 2010_11_02_14_58_42.djvu | Page 32-3]]
Specifically:

[[media:2010_09_02_14_58_46.djvu | Eqn(2) P.6-1]] $$\displaystyle \equiv P_1(x) $$

[[media:2010_09_02_14_58_46.djvu | Eqn(3) P.6-1]] $$\displaystyle \equiv Q_1(x) $$

[[media: 2010_10_28_14_00_19.djvu | p.30-2 ]], Application: Given $$\displaystyle P_1(x), $$ find $$\displaystyle Q_1(x)$$ by variation of parameters

Now...

= Examples: Non-homogeneous L2-ODE-VC =