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...or '''Hill differential equation''' is the second-order linear [[ordinary differential equation]] ...tion is an important example in the understanding of periodic differential equations. Depending on the exact shape of <math> f(t) </math>, solutions may stay bo ...

...Heun|first=Karl L. W. |last=Heun|year=1889}} is the solution of '''Heun's differential equation''' that is holomorphic and 1 at the singular point ''z''&nbsp;=&nb Heun's equation is a second-order [[linear function|linear]] [[ordinary differential equation]] (ODE) of the form ...

...ics, the '''Ince equation''', named for [[Edward Lindsay Ince]], is the [[differential equation]] *{{dlmf|id=28.31|title=Equations of Whittaker–Hill and Ince|first=G. |last=Wolf}} ...

...erential equations|ordinary]] [[complex differential equation|differential equations in the complex plane]] with the '''Painlevé property''' (the only movable s ...are [[pole (complex analysis)|poles]]. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painl ...

...)}}, are solutions {{math|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]: ...)}}, are solutions {{math|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]: ...

...tion''', is a solution of '''Lamé's equation''', a second-order [[ordinary differential equation]]. It was introduced in the paper {{harvs|first=Gabriel|last= Lam ...olutions—called [[periodic instantons]], bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.<ref>H. J. W. Müller-Kirsten ...

...t]]'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common st ...link=Ernst Kummer| last=Kummer |year=1837}}, is a solution to '''Kummer's differential equation'''. This is also known as the confluent hypergeometric function of ...

...</sup>. Multiplying by any function of the form ''nq'' yields more general equations: With ''q''=''d'', these correspond trigonometrically to the equations for the unit circle (<math>x^2+y^2=r^2</math>) and the unit ellipse (<math ...

.... It is a solution of a second-order [[linear function|linear]] [[ordinary differential equation]] (ODE). Every second-order linear ODE with three [[regular singul ...rd|last=Riemann|year=1857}} of the hypergeometric function by means of the differential equation it satisfies. ...

..., sometimes called angular Mathieu functions, are solutions of Mathieu's [[differential equation]] ...cur in problems involving periodic motion, or in the analysis of [[partial differential equation]] [[boundary value problem]]s possessing [[elliptic]]{{Disambiguat ...

...ular, it occurs when solving [[Laplace's equation]] (and related [[partial differential equation]]s) in [[spherical coordinates]]. Associated Legendre polynomials of derivatives of ordinary [[Legendre polynomials]] (''m'' ≥ 0) ...

...ated function '''Bi(''x'')''', are linearly independent solutions to the [[differential equation]] ...' or the '''Stokes equation'''. This is the simplest second-order [[linear differential equation]] with a turning point (a point where the character of the solutio ...

...inary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from [[zonal spherical fun ...asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally. ...

It is related to the ordinary sine integral by ...Integral Taylor series proof |website=Difference Equations to Differential Equations}} ...

...hebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]] ...ebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].) ...

{{Short description|Families of solutions to related differential equations}} ...rich Bessel]], are canonical solutions {{math|''y''(''x'')}} of Bessel's [[differential equation]] ...

...on|Fermi–Dirac]] distributions) and also occurs in the solution of [[delay differential equation]]s, such as {{math|''y''′(''t'') {{=}} ''a'' ''y''(''t'' − 1)}}. I Both authors derived a series solution for their equations. ...

...IXYC&pg=PA28 Extract of page 28]</ref><ref>{{cite book |title=Differential Equations: An Introduction with Mathematica |edition=illustrated |first1=Clay C. |las Other important functional equations for the gamma function are [[reflection formula|Euler's reflection formula] ...

Since the ordinary gamma function is defined as ...h.org/encyclopedia/RigidityTheoremForAnalyticFunctions.html], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In ...