property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE. Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines. Sturm–Liouville theory Sturm–Liouville theory is a theory of a special type of second order linear ordinary differential equation. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined via second-order homogeneous linear equations. The problems are identified as Sturm-Liouville Problems (SLP) and are named after J.C.F. Sturm and J. Liouville, who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations. Existence and uniqueness of solutions There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are {| class="wikitable" |- ! Theorem ! Assumption ! Conclusion |- |Peano existence theorem ||F continuous ||local existence only |- |Picard–Lindelöf theorem ||F Lipschitz continuous ||local existence and uniqueness |- |} In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of Grönwall's inequality are met. Also, uniqueness theorems like the Lipschitz one above do not apply to DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone. Local existence and uniqueness theorem simplified The theorem can be stated simply as follows. For the equation and initial value problem: if F and ∂F/∂y are continuous in a closed rectangle in the x-y plane, where a and b are real (symbolically: ) and denotes the Cartesian product, square brackets denote closed intervals, then there is an interval for some where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F to be linear, this applies to non-linear equations that take the form F(x,