Article ID: CBB000933545

A Historical Survey: Why Did Lagrange Redefine the Complete Integral of a First-Order Partial Differential Equation (2008)

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Yuan, Min (Author)
Jia, Xiaoyong (Author)


Ziran Kexueshi Yanjiu (Studies in the History of Natural Sciences)
Volume: 27
Pages: 485--497


Publication Date: 2008
Edition Details: [Translated title.] In Chinese.
Language: Chinese

Abstract 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 definitionsthe 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.

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Authors & Contributors
Fraser, Craig G.
Ferraro, Giovanni
Patergnani, Elisa
Jouve, Guillame
Mariano Giaquinta
Simonov, N. I.
Concepts
Mathematics
Calculus
Differential and integral equations
Societies; institutions; academies
Differential equations
Statistics
Time Periods
18th century
19th century
17th century
Places
France
Italy
Germany
Europe
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