User:Twine

Week 6, Lecture 3

Constant Coefficient Homogeneous High Order ODEs

Ex Failed to parse (syntax error): {\displaystyle L(y) = ay" + by' +cy = 0, a, b, c \in \mathbb{R}}

Or generally $L(y)=\sum _{k}=0^{n}a_{k}y^{(k)}=0,a_{k}\in \mathbb {R}$

Failed to parse (syntax error): {\displaystyle L:{functions on \mathbb{R} \rightarrow {functions on \mathbb{R}} is a linear transformation ("linear operator").

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

Take Failed to parse (syntax error): {\displaystyle y"+y'-6y = 0} , guess Failed to parse (syntax error): {\displaystyle y = c, y' = \alpha e^{\alpha x}, y" = \alpha^2e^{\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?