The Algebraic Derivative Applied to Laplace’s Differential Equation

Abstract

Pierre Simon Laplace (1749–1827) in his treatise “Théorie analytique des probabilités” of 1817 considered a differential equation which now carries his name and which may be written as

$$ \left( {{a_2}t + {b_2}} \right)y''\left( t \right) + \left( {{a_1}t + {b_1}} \right)y'(t) + \left( {{a_0}t + b} \right)y\left( t \right) = 0, $$

(19.1)

where the a’s and b’s are given complex numbers with a₂≠ 0.