# inhomogeneous equations

(0.002 seconds)

## 1—10 of 20 matching pages

##### 1: 1.13 Differential Equations

…
►

###### §1.13(iii) Inhomogeneous Equations

►The*inhomogeneous*(or*nonhomogeneous*) equation … ►###### Variation of Parameters

…##### 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) Inhomogeneous Equations

… ►A new problem arises, however, if, as $n\to \mathrm{\infty}$, 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 equation
…
►For uniform asymptotic expansions, for large $\nu $ and fixed $\mu =-1,0,1,2,\mathrm{\dots}$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390).
…
…

##### 4: 11.2 Definitions

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

…
##### 8: 11.10 Anger–Weber Functions

##### 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).
…