User:Egm6322.s12.team2.steele.m2/Mtg7

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

[[media:Pea1.f11.mtg7.djvu|Page 7-1]]
Example: Legendre differential operator/equation King 2003 p.31 Two homogeneous solutions:

R*2.1: Verify that $$L_2(y^1_H) = L_2(y^2_H) = 0$$ Note: cf. King 2003 p. 33 $$\{P_n(x), n=0, 1, 2, \ldots\}$$ Legendre polynomials $$\{Q_n(x), n=0,1,2, \ldots\}$$ Legendre functions $$\{[P_n(x), Q_n(x)], n=0,1,2, \ldots\}$$ Legendre functions =Methods of reduction of order=

[[media:Pea1.f11.mtg7.djvu|Page 7-2]]
It may be easier to solve an ODE with lower order than the original ODE with higher order. =Reduction-Of-Order Method 0 (Zero)= Missing dependent variable y(x) Independent variable General nonlinear ODE of order n (Nn-ODE) (without missing y(x)) Nonlinear function Example:

Case of missing dependent variable y(x)

Missing y(x)

[[media:Pea1.f11.mtg7.djvu|Page 7-3]]
Example: See Example p.7-2; delete log y in (2) p.7-2 (1) 						Y(x) missing Reduce order by defining Order (n-1), reduced from order n If (3) can be solved for p(x), then

Integration constant Primitive or antiderivative or indefinite integral

[[media:Pea1.f11.mtg7.djvu|Page 7-4]]
Also		dummy integration variable (2) p.7-3:

Primitive cf. King 2003 p.5 Example: L2-ODE-CC (Constant Coefficients) Y(x) missing

[[media:Pea1.f11.mtg7.djvu|Page 7-5]]
Integration constant is k Note: Method: Euler integration factor, coming soon. For the impatient: See F10 Mtg 7. R*2.2: Verify that (1) is indeed the solution for (6) p.7-4. Solution for (5) p.7-4: =Reduction-Of-Order Method 1: Exact N1-ODEs=

[[media:Pea1.f11.mtg7.djvu|Page 7-6]]
Euler Integrating factor method (IFM) Particular class of N1-ODEs: Linear in $$y\prime$$

$$\frac{dy}{dx}$$

R*2.3: Show that (2) is linear in y’, and that (2) is in general an N1-ODE. But (2) is not the most general N1-ODE as represented by (1). Give an example of a ore general N1-ODE.