Explain how to find the particular solution to each given IVP.
\[ -{x''} - 4 \, {x} = 0 \hspace{2em} {x} (0)= 1 , {x} '(0)= 8 \]
\[ -{y''} - 12 \, {y} = 7 \, {y'} \hspace{2em} {y} (0)= -1 , {y} '(0)= 2 \]
Answer:
\[ {x} = \cos\left(2 \, t\right) + 4 \, \sin\left(2 \, t\right) \]
\[ {y} = -2 \, e^{\left(-3 \, t\right)} + e^{\left(-4 \, t\right)} \]