Scalar Linear 2nd Order ODEs: Obtaining 2nd Solution

This lecture focuses on systematically obtaining a second solution, Y2, to a homogeneous linear second-order ordinary differential equation (ODE), independent of the first solution, Y1. The process involves using Abel's identity and the Wronskian associated with Y1 and Y2. By deriving Abel's identity for a scalar linear second-order ODE, a linear first-order ODE for Y2 is obtained. The lecture explains the steps to solve this first-order ODE and derive an explicit relation for Y2 in terms of Y1. The relation ensures that Y2 is a linearly independent solution to the ODE. Through the superposition principle, it is shown that adding or subtracting a multiple of Y1 to Y2 does not affect its solution status. The lecture concludes by simplifying the relation for Y2 and discussing the flexibility in choosing boundary values.