Answer
The general solution of the given differential equation is
$$
R(t)=e^{-3t}[c_{1}\sin(5t)+c_{2}\cos(5t)].
$$
Work Step by Step
$$
\frac{d^{2} R}{d t^{2}}+6 \frac{d R}{d t}+34 R=0
$$
The given equation is a homogeneous linear equation. So the characteristic equation of the differential equation is
$$
r^{2}+6r+34 =0
$$
whose roots are
$$
\quad \:r_{1,\:2}=\frac{-6\pm \sqrt{6^2-4\cdot \:1\cdot \:34}}{2\cdot \:1}=-3 \pm 5i
$$
Therefore, the general solution of the given differential equation is
$$
R(t)=e^{-3t}[c_{1}\sin(5t)+c_{2}\cos(5t)]
$$
