This chapter explores the Method of Reduction of Order, a powerful analytical technique designed to find a second, linearly independent solution to a second-order linear homogeneous differential equation when one non-trivial solution is already known. The fundamental principle involves assuming a second solution as the product of the known solution and an unknown function; through substitution and differentiation, the original second-order equation is reduced to a more manageable first-order differential equation. Key theoretical foundations discussed include the Wronskian, which serves as a definitive test for the linear independence of solutions, and Abel’s Formula, which provides a direct integral pathway to construct the second solution. Through a series of diverse solved examples—ranging from constant coefficient equations to more complex variable coefficient forms—the chapter demonstrates how to synthesize these components into a general solution. Additionally, the integration of MATLAB examples illustrates how computational tools can verify these analytical results and solve initial value problems efficiently.