User:Egm6321.f10.team4.Yoon/Mtg6

Mtg 6: Thu, 2 Sep 10

= Example (cont'd) =

[[media: 2010_09_02_13_55_50.djvu | Page 6-1]]
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Example [[media: 2010_08_26_13_57_13.djvu | p.5-4]] Continued


 * {| style="width:100%" border="0"

\begin{align} y^1_H(x) = x =: P_1(x) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 *  (2)
 * }
 * }


 * {| style="width:100%" border="0"

\begin{align} y^2_H(x) = \frac{x}{2} \log \left( \frac{1+x}{1-x} \right) - 1 =: Q_1(x) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 *  (3)
 * }
 * }

Next we look at the methods of reduction of order.

= Methods of Reduction of order =

Reduction of order $$\Rightarrow$$ may be easier to solve lower order ODEs

Reduction of order method 0: Missing dependent variable y(x)
$$\displaystyle x$$ = independent variable

$$\displaystyle y(x)$$ = dependent variable

General nonlinear ODE of order n:(without y missing)

[[media: 2010_09_02_14_58_46.djvu | Page 6-2]]
Example:

Missing dependent variable y:

Example:

Reduce order by

If Eqs.(5) can be solved for $$\displaystyle p(x)$$, then

Application: $$\displaystyle y''+y'\underbrace{...}_{\color{blue}{y \ missing}}=x \quad \quad _$$

[[media: 2010_09_02_14_58_46.djvu | Page 6-3]]
Solution:

(Solved by Euler Integrating Factor method see [http://en.wikiversity.org/wiki/User:Egm6321.f09/Lecture_plan) Where A and B are integrating Constants

= Reduction of order method 1: Exact nonlinear 1st-order ODEs =

Integrating Factor Method(Euler)

Nonlinear 1st order ODE (N1_ODE)