Count Riccati and the Early Days of the Riccati Equation

Towards the turn of the seventeenth century, when the baroque was giving way to the enlightenment, there lived in the Republic of Venice a gentleman, the father of nine children, by the name of Jacopo Franceso Riccati. On the cold New Year’s Eve of 1720, he wrote a letter to his friend Giovanni Rizzetti, where he proposed two new differential equations. In modern symbols, these equations can be written as follows: 1.1$$\dot x = \alpha {x^2} + \beta {t^m}$$1.2$$\dot x = \alpha {x^2} + \beta t + \gamma {t^2}$$ where m is a constant. This is probably the first document witnessing the early days of the Riccati Equation, an equation which was to become of paramount importance in the centuries to come.