The method of undetermined coefficients is a powerful technique used to find a particular solution to a non-homogeneous linear differential equation with constant coefficients, i.e., \(a_{n} y^{\left( n \right)} + a_{n - 1} y^{{\left( {n - 1} \right)}} + a_{n - 2} y^{{\left( {n - 2} \right)}} + \ldots + a_{0} y = f\left( x \right)\) . The technique is specifically applicable when the forcing function is a polynomial, exponential, sine, cosine, or a finite combination thereof. The methodology centers on selecting a "trial solution" or "guess" that mirrors the mathematical form of the forcing function, incorporating unknown constants that are subsequently determined by substituting the trial solution into the differential equation and equating coefficients. A critical component of the chapter is the management of Case Duplication, where the initial guess must be modified (typically by multiplying by a power of the independent variable) if it overlaps with the solution to the corresponding homogeneous equation. Through extensive solved examples and MATLAB demonstrations, the chapter illustrates the calculation of complex derivatives, the construction of general solutions, and the verification of results, emphasizing that while particular solutions are not unique, the resulting general solution consistently represents the complete family of solutions.