Question $2$

Consider the differential equation

$y^{\text{'}\text{'}}-4y^{\text{'}}+13y=0$

Two solutions are given $by$

$y_1(t)=e^{2t}sin(3t),y_2(t)=e^{2t}cos(3t)$

Find the solution $y(t)$ satisfying the initial conditions $y(0)=-3,y^{\text{'}}(0)=15.$

$y(t)=7e^{2t}sin(3t)-3e^{2t}cos(3t)$

$y(t)=4e^{2t}sin(3t)-3e^{2t}cos(3t)$

$y(t)=4e^{2t}sin(3t)+3e^{2t}cos(3t)$

$y(t)=-3e^{2t}sin(3t)+7e^{2t}cos(3t)$