\( y ^ { \prime \prime } - 6 y ^ { \prime } + 13 y = 0 \)

To solve the differential equation \( y^{\prime\prime} - 6y^{\prime} + 13y = 0 \), you'll first want to find the characteristic equation by substituting \( y = e^{rt} \), leading to the quadratic equation \( r^2 - 6r + 13 = 0 \). The discriminant \( (-6)^2 - 4 \cdot 1 \cdot 13 = 36 - 52 = -16 \) indicates complex roots. Thus, you find \( r = 3 \pm 2i \), leading to the general solution \( y(t) = e^{3t}(C_1 \cos(2t) + C_2 \sin(2t)) \). This type of second-order linear homogeneous equation often pops up in physics and engineering, particularly in scenarios involving oscillations or wave phenomena. For instance, it can model systems like damped harmonic oscillators or circuits with inductors and capacitors. The resulting solution provides insight into how the system behaves over time, showing exponential growth or decay coupled with periodic behavior!