# particular solutions

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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
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 $\mathbf{E}_{n}\left(1\right)$ is intermediate to those of the complementary functions $J_{n}\left(1\right)$ and $Y_{n}\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. …
##### 5: 32.10 Special Function Solutions
For certain combinations of the parameters, $\mbox{P}_{\mbox{\scriptsize II}}$$\mbox{P}_{\mbox{\scriptsize 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).
##### 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 $I_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
###### §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 $x^{3}+ax+b-y^{2}=0$ (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 $\omega$. … However, in response to a small perturbation at least one solution may become unbounded. …