👉 Hopkins' method, named after mathematician David J. Hopkin, is a powerful tool for analyzing and solving differential equations, particularly those arising in physics and engineering. It involves transforming a given differential equation into a simpler form, often linear and homogeneous, through a change of variables. This transformation is achieved by introducing new independent variables, typically denoted as \( t_1 \) and \( t_2 \), which are related to the original variables \( x \) and \( y \) by specific functions. The key idea is to express the original differential equation in terms of these new variables, where the coefficients become functions of \( t_1 \) and \( t_2 \), simplifying the equation. By solving this new, often separable, equation and then transforming back to the original variables, one can find a solution that satisfies the original differential equation. This method is particularly useful for equations with variable coefficients and boundary conditions, making it a versatile tool in applied mathematics and physics.
