User:EGM6341.s11.TEAM1.WILKS/Mtg27

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

[[media: 2010_10_21_14_05_59.djvu | Mtg 27:]] Thu, 21 Oct 10

= Application: Euler L2-ODE-VC cont'd =

[[media: 2010_10_21_14_05_59.djvu | Page 27-1]]
[[media: 2010_10_19_14_56_03.djvu | Eq.(7)p.26-4]]:

[[media: 2010_10_19_14_56_03.djvu | Eq.(8)p.26-4]]:

Define

and apply the Euler's formula [[media: 2010_10_19_14_56_03.djvu | Eq.(9)p.26-4]].

= Application: Euler L2-ODEs with multiple roots, method of variation of parameters =

HW 5.3

Consider characteristic equation:

where $$ \lambda\ \ $$ is a given number, e.g., $$ \lambda\ =5 \ $$ 1) Euler L2-ODE-VC 1.1) Find $$ a_2, a_1, a_0 \ $$ such that Eq.(4) is characteristic equation of Eq.(5) 1.2) 1st homogeneous solution: $$ y_1=x^{ \lambda\ } \ $$ 1.3) 2nd homogeneous solution: $$ y=c(x)y_1(x) \ $$, where $$ c(x)=u(x) \ $$ which is an unknown.

This is called the Euler method of variation of constants or variation of parameters.

Find the differential equation governing $$ u(x) \ $$.

[[media: 2010_10_21_14_05_59.djvu | Page 27-2]]
Solve for $$ u \ $$ then $$ y_2 \ $$ 1.4) General homogeneous solution 2) Euler L2-ODE-CC Repeat above steps for Eq.(1)

END HW 5.3 HW 5.4

Recall [[media: 2010_09_16_13_50_21.djvu | Page 11-2]], use same idea of variation of constants (parameters) to find the particular solution $$ y_P \ $$ after knowing the homogeneous solution $$ y_H \ $$, i.e., let $$ y(x)=A(x) y_H(x) \ $$.

END HW 5.4

= Application: Nonhomogeneous Euler L2-ODE-CC =

HW 5.5 Non-homogeneous L2-ODE-CC



Equation of motion: 1) Find the partial differential equation(s) governing the integrating factor $$ h(t,y) \ $$.

Reference [[media: 2010_09_23_14_52_54.djvu | Eq.(1)p.15-3]] and [[media: 2010_09_23_14_52_54.djvu | Eq.(2)p.15-3]] (condition 2 of exactness: 2 relationships)

to be continued