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

  1. \[ -{x''} = 12 \, {x} + 7 \, {x'} \hspace{2em} {x} (0)= -3 , {x} '(0)= 10 \]

  2. \[ -{y''} = 16 \, {y} \hspace{2em} {y} (0)= -3 , {y} '(0)= -4 \]

Answer:

\[ {x} = -2 \, e^{\left(-3 \, t\right)} - e^{\left(-4 \, t\right)} \]

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