New Differential and Integral Equations Functions 

Airy's functions solve a homogeneous ODE. The new ScorerGi and ScorerHi functions solve the inhomogeneous version of the same equation.

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X y''[x] - x y[x] /. {{y -> AiryAi}, {y -> AiryBi}, {y -> ScorerHi}, {y -> ScorerGi}} // Simplify

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X Plot[{ScorerHi[x], ScorerGi[x], AiryAi[x], AiryBi[x]}, {x, -5, 2}, PlotLegends -> "Expressions", PlotTheme -> "Detailed", PlotLabel -> Style[Text["Scorer and Airy Functions"], 14, Black, Bold], ImageSize -> Medium]

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LogisticSigmoid solves the logistic ODE .

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X y'[x] == y[x] (1 - y[x]) /. y -> LogisticSigmoid

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X Plot[LogisticSigmoid[x], {x, -6, 6}, ImageSize -> Medium, PlotTheme -> "Business"]

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Decompose the Fresnel integrals FresnelC and FresnelS in terms of the associated Fresnel functions FresnelF and FresnelG.

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X 1/2 + FresnelF[x] Sin[(\[Pi] x^2)/2] - Cos[(\[Pi] x^2)/2] FresnelG[x] // FunctionExpand // Simplify // TraditionalForm

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X 1/2 - Cos[(\[Pi] x^2)/2] FresnelF[x] - FresnelG[x] Sin[(\[Pi] x^2)/2] // FunctionExpand // Simplify // TraditionalForm

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X Plot[{FresnelF[x], FresnelG[x]}, {x, -5, 5}, Filling -> {1 -> {2}}, PlotTheme -> {"Scientific", "Detailed"}, ImageSize -> Medium]

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