aspg-33-4428-1792223338 20261017074858 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 2026 13 1 Hind Kolaib Department of Mathematics, Faculty of Science, University of Tikrit, Iraq This is the complete study of Lagrange equations, a basic formulation in mathematical analysis and classical mechanics. In this report, we present derivation and classification as well as analytical solution techniques for Lagrange type differential equations. These include the Lagrange equations of motion from the calculus of variations, Lagrange multipliers for constrained optimization, and Lagrange interpolating polynomials. All types all begin with the mathematical proof and step to solution algorithms. Several relevant examples are provided to demonstrate the application of these equations in pure & applied mathematics, along with their detailed solutions. 2026 51 56 4428 https://creativecommons.org/licenses/by/4.0/ 10.54216/GJMSA.130104 https://americaspg.com/journal/33/article/4428 https://americaspg.com/storage/articles/manuscripts/01KVNWJ9C5X9WS4DTSTG2NG0RF.pdf