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The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time.
The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin approximation: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a ...
The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation =, known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation d 2 y d x 2 − k y = 0 {\displaystyle {\frac {d^{2}y}{dx^{2}}}-ky=0} is oscillatory for k <0 and exponential for k >0 ...
In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. [2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable , and its specification constitutes the essential or Dirichlet boundary ...
For example, if b = 0 and a ≠ 0 then Γ(a+1)U(a, b, z) − 1 is asymptotic to az ln z as z goes to zero. But see #Special cases for some examples where it is an entire function (polynomial). Note that the solution z 1−b U(a + 1 − b, 2 − b, z) to Kummer's equation is the same as the solution U(a, b, z), see #Kummer's transformation.