Second Order ODEs

Solutions

From a second order LHCC, there are three possible solutions.

Repeated Roots

$r = \alpha\pm 0 \Rightarrow Ae^{\alpha t}+Bte^{\alpha t}$

Distinct Real Roots

$r = \alpha, \beta \Rightarrow Ae^{\alpha t} + Be^{\beta t}$

Imaginary Roots

$r = \alpha\pm\beta i \Rightarrow e^{\alpha t}(A\cos(\beta t) + B\sin(\beta t))$