Twine: Corrected latex errors

2012-10-24T19:33:08Z

Corrected latex errors

← Older revision		Revision as of 15:33, 24 October 2012
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	Constant Coefficient Homogeneous High Order ODEs		Constant Coefficient Homogeneous High Order ODEs

	Ex <math>~~L(y)~~ = ~~ay"~~ + by' +cy = 0, a, b, c \in \mathbb{R}</math>		Ex <math>Ly = a y'' + b y' + c y = 0</math>, <math>a \in \mathbb{R}, b \in \mathbb{R}, c \in \mathbb{R}</math>

	Or generally <math>~~L(y)~~ = \sum_k=0^n a_k y^{(k)} = 0, a_k \in \mathbb{R}</math>		Or generally <math>Ly = \sum_k=0^n a_k y^{(k)} = 0, a_k \in \mathbb{R}</math>

	<math>L:{~~functions~~ on \mathbb{R} \rightarrow {~~functions~~ on \mathbb{R}</math> is a linear transformation ("linear operator").		<math>L:\{f: \mathbb{R} \rightarrow \mathbb{R}\} \rightarrow \{f: \mathbb{R} \rightarrow \mathbb{R}\}</math> is a linear transformation ("linear operator").

	What do we expect from <math>{y:~~L(y)~~ = 0} = ker(L)</math>? We expect an n-dimensional vector space.		What do we expect from <math>\{y: Ly = 0\} = ker(L)</math>? We expect an n-dimensional vector space.

	Take <math> y"+y'-6y = 0</math>, guess <math> y = c, y' = \alpha e^{\alpha x}, y" = \alpha^2e^{\alpha x}</math>		Take <math> y''+y'-6y = 0</math>, guess <math> y = c, y' = \alpha e^{\alpha x}, y'' = \alpha^2e^{\alpha x}</math>

	<math> \alpha^2 e^{\alpha x} + \alpha e^{\alpha x} - 6 e^{\alpha x} = 0</math>		<math> \alpha^2 e^{\alpha x} + \alpha e^{\alpha x} - 6 e^{\alpha x} = 0</math>

Twine: Notes from Oct 19, temporary placement

2012-10-19T14:02:24Z

Notes from Oct 19, temporary placement

New page

Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

Ex <math>L(y) = ay" + by' +cy = 0, a, b, c \in \mathbb{R}</math>

Or generally <math>L(y) = \sum_k=0^n a_k y^{(k)} = 0, a_k \in \mathbb{R}</math>

<math>L:{functions on \mathbb{R} \rightarrow {functions on \mathbb{R}</math> is a linear transformation ("linear operator").

What do we expect from <math>{y:L(y) = 0} = ker(L)</math>? We expect an n-dimensional vector space.

Take <math> y"+y'-6y = 0</math>, guess <math> y = c, y' = \alpha e^{\alpha x}, y" = \alpha^2e^{\alpha x}</math>

<math> \alpha^2 e^{\alpha x} + \alpha e^{\alpha x} - 6 e^{\alpha x} = 0</math>

<math> (\alpha^2 +\alpha - 6) e^{\alpha x} = 0</math>

<math> (\alpha +3)(\alpha - 2) = 0</math>

So we have <math> y = c_1e^{-3x} +c_2 e^{2x}</math> as the general solution.

Say we have complex <math>\alpha</math>. Then what?

User:Twine - Revision history

Twine: Corrected latex errors

Twine: Notes from Oct 19, temporary placement