Kummer's differential equation

The second-order ordinary differential equation x y'' + (c - x) y' - a y = 0, sometimes also called Kummer's differential equation. It has a regular singular point at 0 and an irregular singularity at ∞. The solutions y = b_1 _1 F_1(a;c;x) + b_2 U(a, c, x) are called confluent hypergeometric function of the first and second kinds, respectively. Note that the confluent hypergeometric function of the first kind is also denoted M(a, c, x) or Φ(a;c;z).

confluent hypergeometric function of the first kind | confluent hypergeometric function of the second kind | general confluent hypergeometric differential equation | hypergeometric differential equation | Whittaker differential equation