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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.
In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form ″ + ′ + = has a solution expressible by a generalised Frobenius series when (), () and () are analytic at = or is a regular singular point.
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.
Consider the simple nonlinear second-order differential equation: ″ = (′). This equation is an equation only of y'' and y', meaning it is reducible to the general form described above and is, therefore, separable.
For example, the second order partial derivatives of a scalar function of n variables can be organized into an n by n matrix, the Hessian matrix. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a ...
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 .