## A historical survey: why did Lagrange redefine the complete integral of a first-order partial differential equation.(Chinese. English summary)Zbl 1197.01039

Summary: Euler and Lagrange gave different definitions for the concept of the complete integral respectively in 1770 and 1774. As the foundation of his first-order partial differential equation theory, Lagrange’s definition plays an important role in his general integral theory. Based on a delicate comparison and analysis between the two definitions, the reasons why Lagrange redefined the complete integral of a first-order partial differential equation are revealed. On the one hand, Lagrange hit upon the new idea to redefine the complete integral when he was inspired by his study on the first-order partial differential equation with the method of variation of constants. While he used this new idea to research the ordinary differential equation, he explained the singular integral successfully. Inspired by his method, Lagrange formally put forward his new concept of the complete integral. On the other hand, his new definition showed his spirit of pursuing the general solution of the partial differential equation.

### MSC:

 01A55 History of mathematics in the 19th century 35-03 History of partial differential equations

Lagrange, J. L.