FANDOM



First Order Equation

Given the linear system

$ {\frac{d\hat{x}}{dt}= \hat{A}\hat{x}} $

and eigenvalue and corresponding eigenvector $ {\lambda} $ and $ {\hat{v}} $, then

$ {\hat{x}(t) = \hat{v}e^{\lambda t}} $

is a solution.

Second Order Equation

Given an n x matrix $ {A} $ and associated eigenvalues and eigenvectors $ {-\omega_1^2, -\omega_2^2,..., -\omega_n^2} $ and $ {\hat{v}_1, \hat{v}_2,..., \hat{v}_n} $, the general solution to the equation

$ {\hat{x}'' = \hat{A}\hat{x}} $

is given by

$ \hat{x}(t) = \sum_{i=0}^{n} (c_i cos (\omega_i t) + d_i sin (\omega_i t))\hat{v}_i $