Explain how to find the particular solution to each given IVP.

  1. \[ 0 = -{x''} - 81 \, {x} \hspace{2em} {x} (0)= -3 , {x} '(0)= -45 \]

  2. \[ -20 \, {y} + {y'} = -{y''} \hspace{2em} {y} (0)= -3 , {y} '(0)= -12 \]

Answer:

\[ {x} = -3 \, \cos\left(9 \, t\right) - 5 \, \sin\left(9 \, t\right) \]

\[ {y} = -3 \, e^{\left(4 \, t\right)} \]