# inhomogeneous equations

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##### 1: 1.13 Differential Equations
###### §1.13(iii) InhomogeneousEquations
The inhomogeneous (or nonhomogeneous) equation
##### 2: 3.6 Linear Difference Equations
If $d_{n}=0$, $\forall n$, then the difference equation is homogeneous; otherwise it is inhomogeneous. …
###### §3.6(iv) InhomogeneousEquations
A new problem arises, however, if, as $n\to\infty$, the asymptotic behavior of $w_{n}$ is intermediate to those of two independent solutions $f_{n}$ and $g_{n}$ of the corresponding inhomogeneous equation (the complementary functions). … 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. … or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …
##### 3: 11.9 Lommel Functions
The inhomogeneous Bessel differential equationFor uniform asymptotic expansions, for large $\nu$ and fixed $\mu=-1,0,1,2,\dots$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
##### 5: 3.7 Ordinary Differential Equations
If $h=0$ the differential equation is homogeneous, otherwise it is inhomogeneous. … … (This can happen only for inhomogeneous equations.) … The latter is especially useful if the endpoint $b$ of $\mathscr{P}$ is at $\infty$, or if the differential equation is inhomogeneous. …
##### 6: 11.13 Methods of Computation
A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … Sequences of values of $\mathbf{H}_{\nu}\left(z\right)$ and $\mathbf{L}_{\nu}\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …
##### 7: 10.15 Derivatives with Respect to Order
10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
##### 8: 11.10 Anger–Weber Functions
The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation
##### 9: 2.8 Differential Equations with a Parameter
For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). …
##### 10: 14.29 Generalizations
For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).