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Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Numerov's method (also called Cowell's method) is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear multistep method. The method is implicit, but can be made explicit if the differential equation is linear.
Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y 1 ( x ) {\displaystyle y_{1}(x)} is known and a second linearly independent solution y 2 ( x ) {\displaystyle y_{2}(x)} is desired.
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.
A method of matched asymptotic expansions - with matching of solutions in the common domain of validity - has been developed and used extensively by Dingle and Müller-Kirsten for the derivation of asymptotic expansions of the solutions and characteristic numbers (band boundaries) of Schrödinger-like second-order differential equations with ...
In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for ...
Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if a {\displaystyle a} is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic.
As a second-order differential operator, the Laplace operator maps C k functions to C k−2 functions for k ≥ 2.It is a linear operator Δ : C k (R n) → C k−2 (R n), or more generally, an operator Δ : C k (Ω) → C k−2 (Ω) for any open set Ω ⊆ R n.