This chapter examines the theory and application of homogeneous linear differential equations with constant coefficients. The core of the discussion centers on the construction of the characteristic equation and the analysis of its roots—whether real and distinct, repeated, or complex conjugates—to determine the structure of the general solution. Through a variety of examples ranging from first-order to high-order equations, the text demonstrates how these solutions form a vector space where the general solution is expressed as a linear combination of linearly independent functions, such as exponentials, polynomials, and trigonometric or hyperbolic functions. The chapter also provides a detailed guide on solving initial value problems by determining unique constants through systems of linear equations. Furthermore, the integration of MATLAB is emphasized as a vital tool for calculating high-order polynomial roots, verifying symbolic solutions, and simplifying complex algebraic results, providing a robust bridge between theoretical derivation and computational verification.