User:Egm6322.s12.team2.steele.m2/Mtg35

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

[[media:Pea1.f11.mtg35.djvu|Page 35-0]]
Plan: $$\Rightarrow$$Variation of parameters: Reduction-of-order method 2 cont’d Comparison with the method in King 2003 p.8 Application: Legendre equation with n=1 Application: King 2003 p.28 (Pb.1.1 ab)

[[media:Pea1.f11.mtg35.djvu|Page 35-1]]
=Comparison with the Method in King 2003 p.8= R*6.2: Show that [[media:Pea1.f11.mtg34.djvu|Eqns(1) p.34-6]] agrees with King 2003 p.8 (1.6), i.e. Being the Wronskian (test of linear independence of $$u_1$$ and $$u_2$$ )

Discuss the feasibility of the following choices for variation of parameters:

[[media:Pea1.f11.mtg35.djvu|Page 35-2]]
Application: Legendre equation with n=1 $$(1-x^2)y\prime\prime-2xy\prime + 2y = 0$$					(2) p.7-1[[media:Pea1.f11.mtg7.djvu|Eqn(2) p.7-1]] Given: $$u_1(x) = x$$								(3) p.7-1 Show: $$u_2(x) = \frac{x}{2} \log \frac{1+x}{1-x} -1$$				(4) p.7-1

Note [[media:Pea1.f11.mtg34.djvu|Eqn(4) p.34-5]]

[[media:Pea1.f11.mtg35.djvu|Page 35-3]]
$$=x \int \left[ \frac{1}{x^2} + \frac{1}{2(1+x)} + \frac{1}{2(1-x)} \right] \, dx$$ = (4) p.7-1 Application: King 2003 p.28 (Pb.1.1 ab) 1st homogeneous solution; pretend not knowing Trial solution: $$y = e^{rx}$$, r = constant	(4) p.30-4 Characteristic equation:

[[media:Pea1.f11.mtg35.djvu|Page 35-4]]
R*6.3: Explain why $$r_2(x)$$ is not a valid root, i.e., $$u_2(x) = e^{xr_2(x)}$$ R*6.4: For the L2-ODE-VC (1) p.35-3, select a valid homogeneous solution, and call it $$u_1$$. Find the 2nd homogeneous solution $$u_2(x)$$ by variation of parameters, and compare to $$e^{xr_2(x)}$$. See also F09 p.21-4.