1. The problem is to solve a second-order ordinary differential equation (ODE). 2. A common form of a second-order ODE is $$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$$ where $a$, $b$, and $c$ are constants and $f(x)$ is a function of $x$. 3. To solve, first find the complementary solution by solving the homogeneous equation $$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$$. 4. The characteristic equation is $$ar^2 + br + c = 0$$. 5. Solve for roots $r$ using the quadratic formula $$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 6. Depending on the discriminant $\Delta = b^2 - 4ac$, the solution form varies: - If $\Delta > 0$, two distinct real roots $r_1$ and $r_2$, solution is $$y_c = C_1 e^{r_1 x} + C_2 e^{r_2 x}$$. - If $\Delta = 0$, one repeated root $r$, solution is $$y_c = (C_1 + C_2 x) e^{r x}$$. - If $\Delta < 0$, complex roots $\alpha \pm \beta i$, solution is $$y_c = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))$$. 7. Next, find a particular solution $y_p$ depending on $f(x)$ using methods like undetermined coefficients or variation of parameters. 8. The general solution is $$y = y_c + y_p$$. 9. Apply initial or boundary conditions to solve for constants $C_1$ and $C_2$. This is the general approach to solving second-order ODEs with constant coefficients.