User:Twine

Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

Ex [math]\displaystyle{ Ly = a y'' + b y' + c y = 0 }[/math], [math]\displaystyle{ a \in \mathbb{R}, b \in \mathbb{R}, c \in \mathbb{R} }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ \{y: Ly = 0\} = ker(L) }[/math]? We expect an n-dimensional vector space.

Take [math]\displaystyle{ y''+y'-6y = 0 }[/math], guess [math]\displaystyle{ y = c, y' = \alpha e^{\alpha x}, y'' = \alpha^2e^{\alpha x} }[/math]

[math]\displaystyle{ \alpha^2 e^{\alpha x} + \alpha e^{\alpha x} - 6 e^{\alpha x} = 0 }[/math]

[math]\displaystyle{ (\alpha^2 +\alpha - 6) e^{\alpha x} = 0 }[/math]

[math]\displaystyle{ (\alpha +3)(\alpha - 2) = 0 }[/math]

So we have [math]\displaystyle{ y = c_1e^{-3x} +c_2 e^{2x} }[/math] as the general solution.

Say we have complex [math]\displaystyle{ \alpha }[/math]. Then what?