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

… ►

9.12.2 $$w(z)=A{w}_1(z)+B{w}_2(z)+p(z),$$

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Defines:

$w$: function (locally)

Symbols:

$z$: complex variable, $A$: constant, $B$: constant, $w_1$: function, $w_2$: function and $p$: particular solution

Permalink:

http://dlmf.nist.gov/9.12.E2

Encodings:

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See also:

Annotations for §9.12(i), §9.12 and Ch.9

►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: 32.10 Special Function Solutions

… ►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. …

5: 3.6 Linear Difference Equations

… ►Thus the asymptotic behavior of the particular solution ${𝐄}_n\left(1\right)$ 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. …

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

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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 ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, 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

… ►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

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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. …