Comparison: Variation of Parameters vs. Undetermined Coefficients
When solving non-homogeneous linear differential equations, finding the particular solution (often written as y_p) is the most critical step. You generally have two tools in your belt: the Method of Undetermined Coefficients and the Method of Variation of Parameters. While both aim for the same result, they operate on very different logic. Here is a breakdown of how they stack up. 1. Undetermined Coefficients: The "Educated Guess" This method is the "shortcut" of the differential equation world. It relies on the fact that for certain types of functions, the derivative looks remarkably like the original function. * How it works: You assume y_p takes the same form as the non-homogeneous term g(x). If g(x) is an exponential like e^(2x), you guess A * e^(2x). If g(x) is sin(x), you guess A * cos(x) + B * sin(x). * The Constraint: It only works if g(x) is a polynomial, exponential, sine/cosine, or a product/sum of these. * The Big Pro: It is algebraically straightforw...