# homogeneous equations

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## 1—10 of 29 matching pages

##### 1: 3.6 Linear Difference Equations

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###### §3.6(ii) Homogeneous Equations

… ►Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. … ►See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … ►If, as $n\to \mathrm{\infty}$, the wanted solution ${w}_{n}$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution ${w}_{n}$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. …##### 2: 3.7 Ordinary Differential Equations

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►If $h=0$ the differential equation is

*homogeneous*, otherwise it is*inhomogeneous*. … … ►The equations can then be solved by the method of §3.2(ii), if the differential equation is homogeneous, or by Olver’s algorithm (§3.6(v)). …##### 3: 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). …##### 4: 10.15 Derivatives with Respect to Order

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10.15.1
$$\frac{\partial {J}_{\pm \nu}\left(z\right)}{\partial \nu}=\pm {J}_{\pm \nu}\left(z\right)\mathrm{ln}\left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu}\sum _{k=0}^{\mathrm{\infty}}{(-1)}^{k}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma}\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^{2})}^{k}}{k!},$$

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10.15.3
$${\frac{\partial {J}_{\nu}\left(z\right)}{\partial \nu}|}_{\nu =n}=\frac{\pi}{2}{Y}_{n}\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^{n}}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^{k}{J}_{k}\left(z\right)}{k!(n-k)}.$$

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10.15.4
$${\frac{\partial {Y}_{\nu}\left(z\right)}{\partial \nu}|}_{\nu =n}=-\frac{\pi}{2}{J}_{n}\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^{n}}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^{k}{Y}_{k}\left(z\right)}{k!(n-k)},$$

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10.15.5
$${\frac{\partial {J}_{\nu}\left(z\right)}{\partial \nu}|}_{\nu =0}=\frac{\pi}{2}{Y}_{0}\left(z\right),{\frac{\partial {Y}_{\nu}\left(z\right)}{\partial \nu}|}_{\nu =0}=-\frac{\pi}{2}{J}_{0}\left(z\right).$$

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10.15.6
$${\frac{\partial {J}_{\nu}\left(x\right)}{\partial \nu}|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\mathrm{sin}x-\mathrm{Si}\left(2x\right)\mathrm{cos}x\right),$$

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##### 5: 15.11 Riemann’s Differential Equation

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►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1).
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►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by
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##### 6: 31.14 General Fuchsian Equation

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►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions).
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##### 7: 2.9 Difference Equations

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

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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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##### 9: 31.2 Differential Equations

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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\u2102\cup \{\mathrm{\infty}\}$, can be transformed into (31.2.1).
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