Virtual Lessons: Reduction of Order

Reduction of order is a technique used to solve second-order linear homogeneous differential equations, particularly those with constant coefficients. The process involves using the known solution to the differential equation, also known as the homogeneous solution, to find a second solution that is linearly independent of the first solution.

To perform reduction of order, one first finds the homogeneous solution of the differential equation. Then, a second solution is assumed in the form of the product of the homogeneous solution and a function of x, known as the unknown function. The derivative of the assumed solution is then substituted back into the original differential equation, and terms are simplified.

This results in a second-order linear homogeneous differential equation involving only the unknown function and its derivatives. By solving this equation for the unknown function, a second linearly independent solution is obtained. Finally, the general solution to the differential equation is obtained by taking a linear combination of the two linearly independent solutions.