Linear Differential Equations: Second Order with Constant Coefficients

This lecture discusses the particular case of linear differential equations of the second order with constant coefficients. The instructor begins by recalling the general solution methods for first-order equations and transitions to second-order equations. The focus is on deriving solutions for homogeneous equations without a second member, using exponential functions. The lecture details the characteristic equation derived from the differential equation, exploring three cases based on the discriminant: positive, zero, and negative. For each case, the instructor explains how to find the general solution, emphasizing the importance of linear independence among solutions. The discussion includes the derivation of solutions using exponential functions for positive discriminants, a double root scenario for zero discriminants, and complex solutions for negative discriminants. The lecture concludes with a summary of how to solve second-order linear differential equations with constant coefficients, setting the stage for future examples and applications involving non-homogeneous equations.