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1: 11.2 Definitions
Particular solutions: … Particular solutions: …
2: Frank W. J. Olver
, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …
3: 9.12 Scorer Functions
9.12.2 w ( z ) = A w 1 ( z ) + B w 2 ( z ) + p ( z ) ,
where A and B are arbitrary constants, w 1 ( z ) and w 2 ( z ) are any two linearly independent solutions of Airy’s equation (9.2.1), and p ( z ) is any particular solution of (9.12.1). Standard particular solutions are …
4: 3.6 Linear Difference Equations
Thus the asymptotic behavior of the particular solution E n ( 1 ) is intermediate to those of the complementary functions J n ( 1 ) and Y n ( 1 ) ; moreover, the conditions for Olver’s algorithm are satisfied. …
5: 32.10 Special Function Solutions
For certain combinations of the parameters, P II P VI  have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. …
6: 15.11 Riemann’s Differential Equation
The most general form is given by … The complete set of solutions of (15.11.1) is denoted by Riemann’s P -symbol: …In particular, …denotes the set of solutions of (15.10.1).
§15.11(ii) Transformation Formulas
7: 18.38 Mathematical Applications
However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. …
8: 10.25 Definitions
Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. …
§10.25(ii) Standard Solutions
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . …
§10.25(iii) Numerically Satisfactory Pairs of Solutions
Table 10.25.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.25.1). …
9: 23.20 Mathematical Applications
Values of x are then found as integer solutions of x 3 + a x + b - y 2 = 0 (in particular x must be a divisor of b - y 2 ). …
10: 28.33 Physical Applications
  • Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

  • The wave equation …The general solution of the problem is a superposition of the separated solutions. … In particular, the equation is stable for all sufficiently large values of ω . … However, in response to a small perturbation at least one solution may become unbounded. …