Explore four methods for solving second-order ordinary differential equations in this 37-minute video lecture on the harmonic oscillator. Begin with deriving the spring-mass equations from Newton's Second Law, then progress through guessing the solution, using Taylor Series, guessing an alternative form, and writing as a matrix system of equations. Gain a comprehensive understanding of solving the mass-on-spring problem through multiple approaches, enhancing your skills in differential equations and mathematical problem-solving. Follow along as the instructor breaks down each method step-by-step, providing valuable insights into the application of these techniques in physics and engineering.