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inhomogeneous equations

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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 , n , then the difference equation is homogeneous; otherwise it is inhomogeneous. …
§3.6(iv) Inhomogeneous Equations
A new problem arises, however, if, as n , 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 ν and fixed μ = - 1 , 0 , 1 , 2 , , 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
§11.2(ii) Differential Equations
Modified Struve’s Equation
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 𝒫 is at , 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 H ν ( z ) and L ν ( z ) , 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 J ± ν ( z ) ν = ± J ± ν ( z ) ln ( 1 2 z ) ( 1 2 z ) ± ν k = 0 ( - 1 ) k ψ ( k + 1 ± ν ) Γ ( k + 1 ± ν ) ( 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 ξ 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).