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

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