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

  1. \[ 36 \, {x} = -{x''} \hspace{2em} {x} (0)= 1 , {x} '(0)= -12 \]

  2. \[ 0 = 10 \, {y} + {y''} - 7 \, {y'} \hspace{2em} {y} (0)= -2 , {y} '(0)= -13 \]

Answer:

\[ {x} = \cos\left(6 \, t\right) - 2 \, \sin\left(6 \, t\right) \]

\[ {y} = -3 \, e^{\left(5 \, t\right)} + e^{\left(2 \, t\right)} \]