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11: 19.18 Derivatives and Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). ►More concisely, if $v=R_{-a}(𝐛;𝐳)$, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: …

12: 1.13 Differential Equations

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§1.13(iii) Inhomogeneous Equations

… ►If $w_0(z)$ is any one solution, and $w_1(z)$, $w_2(z)$ are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as … ►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). …

13: 23.1 Special Notation

… ►Walker (1996) normalizes $2{\omega }_1=1$, $2{\omega }_3=\tau$, and uses homogeneity (§23.10(iv)). …

14: 2.9 Difference Equations

… ►or equivalently the second-order homogeneous linear difference equation … ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). …

15: 14.30 Spherical and Spheroidal Harmonics

… ►In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). …

16: 19.19 Taylor and Related Series

… ►For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial …

17: 28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. …

18: Bibliography G

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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:

ISSN 0036-1429, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(v).

Encodings:

BibTeX

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Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89

External Links:

Document, MathReview (A. Korányi), ZentralBlatt

Referenced by:

§35.3(i).

Encodings:

BibTeX

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19: 3.8 Nonlinear Equations

… ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …

20: Bibliography F

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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External Links:

ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

BibTeX

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