User:Twine

Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

Ex $Ly=ay''+by'+cy=0$, $a\in \mathbb {R} ,b\in \mathbb {R} ,c\in \mathbb {R}$

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 $y=c,y'=\alpha e^{\alpha x},y''=\alpha ^{2}e^{\alpha x}$

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

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

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

So we have $y=c_{1}e^{-3x}+c_{2}e^{2x}$ as the general solution.

Say we have complex $\alpha$. Then what?