text: In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow. It is
  Here,  is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have , where  is the specific heat ratio and  is the stagnation enthalpy, in which case the Chaplygin's equation reduces to
  The Bernoulli equation (see the derivation below) states that maximum velocity occurs when specific enthalpy is at the smallest value possible; one can take the specific enthalpy to be zero corresponding to absolute zero temperature as the reference value, in which case  is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions.