User:Egm6322.s12.team2.steele.m2/Mtg31

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

[[media:Pea1.f11.mtg31.djvu|Page 31-0]]
Plan: Application: Euler L2-ODE-VC p.30-7 cont’d Application: Euler L3-ODE-VC

[[media:Pea1.f11.mtg31.djvu|Page 31-1]]
=Application: Euler L2-ODE-VC p.30-7 cont’d= See F09 Mtg 26 for higher-order derivatives:

R*5.4: Find $$y_{xxxxx}$$ in terms of the derivatives of y with respect to t. Recall the Euler L2-ODE-VC [[media:Pea1.f11.mtg30.djvu|Eqn(1) p.30-5:]] $$x^2 y\prime\prime-2xy\prime + 2y = 0$$					(1) p.30-5 Use [[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-5]] and (2) p.30-7 in (1) p.30-5 to obtain the Euler L2-ODE-CC:

[[media:Pea1.f11.mtg31.djvu|Page 31-2]]
Stage 2: Trial solution $$e^{rt}$$

Note (1) and (3) p.31-1 leads to the characteristic equation:

So the solution is: Note [[media:Pea1.f11.mtg30.djvu|Eqns(2)-(3) p.30-5:]]

[[media:Pea1.f11.mtg31.djvu|Page 31-3]]
R*5.5: Solve (1) p.30.5 using Method 2 with trial solution: $$y = x^r$$	[[media:Pea1.f11.mtg30.djvu|Eqn(5) p.30-4]] And with boundary conditions:

Plot the solution. Application: Euler L3-ODE-VC

Method 2: Trial solution (5) p.30-4

[[media:Pea1.f11.mtg31.djvu|Page 31-4]]
$$y = x^{4}, \ {\color{red} r = \text{constant}}$$	(5) p.30-4

Characteristic equation:

$$i := \sqrt{-1}$$	[[media:Pea1.f11.mtg27.djvu|Eqn(2) p.27-6]] Hence:

[[media:Pea1.f11.mtg31.djvu|Page 31-5]]
=Euler’s formula: = $$e^{ikx} = \cos kx + i \sin kx$$	[[media:Pea1.f11.mtg27.djvu|Eqn(5) p.27-6]]

R5.6: Show that the trial solution (5) p.30-4 in Method 2 is equivalent to the combined trial solutions [[media:Pea1.f11.mtg30.djvu|Eqns(3)-(4) p.30-4]] in Method 1.