User:Egm6321.f10.team4.Yoon/Mtg28

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

[[media: 2010_10_21_14_48_38.djvu | Mtg 28:]] Thu, 21 Oct 10

= Application: Nonhomogeneous L2-ODE-CC cont'd =

[[media: 2010_10_21_14_48_38.djvu | Page 28-1]]
Recall:

Undergraduate; solve [[media: 2010_10_21_14_05_59.djvu | Eqn(2) p.27-2]] 1) Find $$\displaystyle y_H$$ (Homogeneous Solutions) 2) Find $$\displaystyle y_p$$ (Particular Solutions) Use look-up table (F09 Mtgs [[media: Egm6321.f09.mtg20.djvu | 20]], [[media: Egm6321.f09.mtg22.djvu | 22]]) with a few Particular cases

HW 5.5: Special IFM to solve for general f(t) in [[media: 2010_10_21_14_05_59.djvu | Eqn(2) p.27-2]].

End Recall

{| style="width:100%" border="0" align="left" HW 5.5: cont'd
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2) Trial Solution for integrating factor

which is similar to trial solution for Euler L2-ODE-CC (a homogeneous Ln-ODE-CC), where $$\displaystyle \alpha $$ is unknown to be determined.

[[media: 2010_10_21_14_48_38.djvu | Page 28-2]]
Assume $$\displaystyle \bar a_2 = 0$$ to reduce the order of the resulting ODE so that

2.1) Find $$\displaystyle \bar a_1 $$ and $$\displaystyle \bar a_0 $$in terms of $$\displaystyle a_2, a_1, a_0$$.

Hint: Differentiate the r.h.s. of (1) and compare to integrand on l.h.s. of (1); see also [[media: 2010_10_21_14_48_38.djvu | Eqn(2) p.28-1]].

2.2) Find quadratic equation for $$\displaystyle \alpha $$

Hint: Use $$\displaystyle (2)_2$$ & $$\displaystyle (2)_3$$

2.3) Reduced-order equation: [[media: 2010_10_21_14_48_38.djvu | Eqn(1),(2)p.28-1]] lead to

which is an L1-ODE-CC easily solvable by the IFM.

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2.4) Use the IFM.

[[media: 2010_09_14_15_00_52.djvu | Eqn(1)p.10-3:]]

$$\displaystyle h(t) = \exp [ \int \underbrace{\frac{\bar a_0}{\bar a_1}}_{\displaystyle \beta} dt ] = e^{\beta t} $$

Find y (for general excitation f(t))

2.5) Show that $$\displaystyle \alpha \beta = \frac{a_0}{a_2}$$ and $$\displaystyle \alpha + \beta = \frac{a_1}{a_2}$$, thus $$\displaystyle (\alpha, \beta)$$ are roots of the quadratic equation:

$$\displaystyle (\lambda - \alpha)(\lambda - \beta) = \lambda^2 - \underbrace{(\alpha + \beta)}_{\displaystyle \frac{a_1}{a_2}}\lambda + \underbrace{\alpha \beta}_{\displaystyle\frac{a_0}{a_2}} = 0$$

which is the same as [[media: 2010_10_21_14_48_38.djvu | Eqn(3) p.28-2.]]

2.6) Deduce the expression for the particular Solution $$\displaystyle y_P $$ for a general excitation f(t).

2.7) Verify with table of particular solutions consider $$\displaystyle f(t)=t^2 $$


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Next the method of variation of parameters.

= Variation of parameters: Reduction-of-order method 2 =

[[media: 2010_10_21_14_48_38.djvu | Page 28-4]]
Variation of parameters: a reduction-of-order method History: (Battin 1999 Astrodynamics) Euler 1748, 1752: Mutual perturbations of Jupiter & Saturn Variation of orbital elements variation of constants Lagrange 1782: General variation of parameters perturbations comets on elliptic orbits Dramatic success: Adams(1819 - 1892), LeVerrier (1811-1877) Discovery of Neptune based on irregular motion of Uranus. Neptune: observed (after prediction) by Galle 23 Sep 1846

Important homogeneous L2-ODE-VC
Homogeneous L2-ODE-VC (e.g. Legendre equation, Bessel equation, ...) Given one homogeneous solution $$\displaystyle u_1(x)$$ Find 2nd homogeneous solution $$\displaystyle u_2(x)$$ such that

Assume full homogeneous solution of the form $$\displaystyle y(x) = U(x) u_1(x)$$, where $$\displaystyle U(x)$$ is the unknown function to be found.