User:Twine

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Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

Ex Ly = a y'' + b y' + c y = 0, a \in \mathbb{R}, b \in \mathbb{R}, c \in \mathbb{R}

Or generally Ly = \sum_k=0^n a_k y^{(k)} = 0, a_k \in \mathbb{R}

L:\{f: \mathbb{R} \rightarrow \mathbb{R}\} \rightarrow \{f: \mathbb{R} \rightarrow \mathbb{R}\} is a linear transformation ("linear operator").

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

Take  y''+y'-6y = 0, guess  y = c, y' = \alpha e^{\alpha x}, y'' = \alpha^2e^{\alpha x}

 \alpha^2 e^{\alpha x} + \alpha e^{\alpha x} - 6 e^{\alpha x} = 0

 (\alpha^2 +\alpha - 6) e^{\alpha x} = 0

 (\alpha +3)(\alpha - 2) = 0

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

Say we have complex \alpha. Then what?