$$Question:\, \, Find \, \, the\, \, general \, \, solution\, \, of\, \, the\, \, given\, \, differential\, \, equation:{y}''+2{y}'+2y=0\\\\\\To \, \, begin\, \, with,let\, \, y=e^{rt},{y}'=re^{rt},{y}''=r^2e^{rt}\\\\\\\\Then, r^2+2r+2=0, \, \, r1=\frac{-2+2i}{2}=-1+i,\, \, r2=\frac{-2-2i}{2}=-1-i\\\\\\\\Substitute \, \, \lambda =-1\, \, and \, \, \mu =1\, \, in \, \,\, \, y=C1e^{\lambda t}cos(\mu t)+C2e^{\lambda t}sin(\mu t):\\\\\\\\y=C1e^{-t}cost(t)+C2e^{-t}sin(t)$$
