User:Twine

Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

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

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

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

What do we expect from [math]\displaystyle{ {y:L(y) = 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?