# particular solutions

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##### 1: 11.2 Definitions

##### 2: Frank W. J. Olver

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►, 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.
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##### 3: 9.12 Scorer Functions

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►

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
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##### 4: 3.6 Linear Difference Equations

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►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.
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##### 5: 32.10 Special Function Solutions

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►For certain combinations of the parameters, ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ 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.
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##### 6: 15.11 Riemann’s Differential Equation

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►The most general form is given by
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►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

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►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.
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##### 8: 10.25 Definitions

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►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 ${(\frac{1}{2}z)}^{\nu}$, is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►###### §10.25(iii) Numerically Satisfactory Pairs of Solutions

►Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). …##### 9: 23.20 Mathematical Applications

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►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 $b-{y}^{2}$).
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