1. **State the problem:** Solve the second-order linear differential equation $$\frac{d^2 x}{dt^2} - \frac{dx}{dt} - 6x = 0.$$\n\n2. **Identify the type of equation:** This is a homogeneous linear differential equation with constant coefficients.\n\n3. **Write the characteristic equation:** Replace $$\frac{d^2 x}{dt^2}$$ by $$r^2$$, $$\frac{dx}{dt}$$ by $$r$$, and $$x$$ by 1 to get:\n$$r^2 - r - 6 = 0.$$\n\n4. **Solve the characteristic equation:** Factor or use the quadratic formula. Factoring:\n$$r^2 - r - 6 = (r - 3)(r + 2) = 0,$$\nso the roots are $$r = 3$$ and $$r = -2.$$\n\n5. **Write the general solution:** Since the roots are real and distinct, the general solution is:\n$$x(t) = C_1 e^{3t} + C_2 e^{-2t},$$\nwhere $$C_1$$ and $$C_2$$ are arbitrary constants determined by initial conditions.