1. The problem: Understand the method of reduction of order in differential equations. 2. Reduction of order is a technique used to find a second, linearly independent solution to a second-order linear differential equation when one solution $y_1$ is already known. 3. The general form of a second-order linear differential equation is: $$y'' + p(x)y' + q(x)y = 0$$ 4. If one solution $y_1(x)$ is known, the goal is to find a second solution $y_2(x)$ of the form: $$y_2 = v(x) y_1(x)$$ where $v(x)$ is an unknown function to be determined. 5. Substitute $y_2 = v y_1$ into the original differential equation and use the fact that $y_1$ satisfies the equation to simplify. 6. This substitution reduces the order of the differential equation for $v(x)$ from second order to first order, hence the name "reduction of order." 7. After simplification, you solve for $v'(x)$ and then integrate to find $v(x)$. 8. Finally, the second solution is: $$y_2 = y_1(x) \int \frac{e^{-\int p(x) dx}}{y_1(x)^2} dx$$ 9. This method is very useful when the first solution is known but the second is not easily found by other methods. 10. Summary: Reduction of order helps find a second solution to a second-order linear ODE using a known solution by reducing the problem to a first-order equation for an auxiliary function $v(x)$.