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

  1. \[ {x''} - 2 \, {x'} = 3 \, {x} \hspace{2em} {x} (0)= -1 , {x} '(0)= -19 \]

  2. \[ 0 = -25 \, {y} - {y''} \hspace{2em} {y} (0)= -4 , {y} '(0)= 5 \]

Answer:

\[ {x} = -5 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)} \]

\[ {y} = -4 \, \cos\left(5 \, t\right) + \sin\left(5 \, t\right) \]