1. **Problem statement:** Given one solution of a second-order linear differential equation, find the second solution using the reduction of order formula. 2. **Formula:** If $y_1$ is a known solution of the differential equation $$y'' + p(x)y' + q(x)y = 0,$$ then the second solution $y_2$ can be found by $$y_2 = y_1(x) \int \frac{e^{-\int p(x) \, dx}}{y_1(x)^2} \, dx.$$ 3. **Explanation:** This formula uses the known solution $y_1$ to reduce the order of the differential equation and find a second, linearly independent solution $y_2$. 4. **Steps:** - Compute the integral $\int p(x) \, dx$. - Calculate the exponential $e^{-\int p(x) \, dx}$. - Form the integrand $\frac{e^{-\int p(x) \, dx}}{y_1(x)^2}$. - Integrate this expression with respect to $x$. - Multiply the result by $y_1(x)$ to get $y_2$. 5. **Result:** The function $y_2$ obtained is the second solution to the differential equation, linearly independent from $y_1$. This method is essential when the differential equation cannot be solved by simpler methods and one solution is already known.