Inverse differential equations

This article discusses the concept of inverse differential equations, where instead of solving a differential equation to find functions, you analyze the equation to understand the functions it defines. It uses Bessel functions as a key example, showing how their defining differential equation, x²y″ + xy′ + (x² − n²)y = 0, can be inspected to determine properties like analyticity, singularity, and asymptotic behavior. The author explains the indicial equation method and how it reveals two solutions (Jn and Yn) with distinct behaviors near zero and infinity. This approach is valuable for users of special functions who may not have encountered their differential origins, offering deeper insights without full solving.