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

To solve the differential equation \( y^{\prime\prime} - 6y^{\prime} + 13y = 0 \), we first form the characteristic equation by substituting \( y = e^{rt} \). This yields \( r^2 - 6r + 13 = 0 \). Solving for \( r \) using the quadratic formula gives us complex roots \( r = 3 \pm 2i \). Hence, the general solution to the differential equation can be expressed as \( y(t) = e^{3t}(C_1 \cos(2t) + C_2 \sin(2t)) \), where \( C_1 \) and \( C_2 \) are constants determined by initial conditions. Think of this equation like a rollercoaster ride where the imaginary part \( 2i \) contributes to the oscillations while the real part \( 3 \) ensures a steady upward trend! You can visualize how these curves undulate, giving you a complex yet fascinating motion over time. If you find this thrilling, you might want to explore other second-order linear systems—they can get wild and wacky!