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RESOLUCI?N DE LA ECUACI?N DE AIRYConstruimos la ecuaci?n diferencial en estudio 

> ecuacion:=diff(y(x),x$2)-x*y(x)=0;
y(x)=1/sqrt(Pi)*Int(cos(x*z+z^3/3),z=0..infinity);
 

`:=`(ecuacion, `+`(diff(y(x), `$`(x, 2)), `-`(`*`(x, `*`(y(x))))) = 0)
 

y(x) = `/`(`*`(Int(cos(`+`(`*`(x, `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 3))))), z = 0 .. infinity)), `*`(`^`(Pi, `/`(1, 2))))
 

La soluci?n general ser?: 

> dsolve(ecuacion,y(x));
 

y(x) = `+`(`*`(_C1, `*`(AiryAi(x))), `*`(_C2, `*`(AiryBi(x))))
 

Para graficar la soluci?n suponemos C1 = C2 = 1 

> plot((AiryAi(x)+AiryBi(x)),x=-3..3);
 

Plot_2d
 

Las funciones de onda  AiryAi(x) y AiryBi(x) son soluciones linealmente independientes para la ecuaci?n dada. 

Veamos como se define la funcion AiryAi(x): 

> FunctionAdvisor(definition,AiryAi(x));plot(AiryAi(x),x=-3..3);
 

 

[AiryAi(x) = `+`(`/`(`*`(`/`(1, 6), `*`(`+`(`-`(`/`(`*`(3, `*`(x, `*`(`^`(3, `/`(1, 6)), `*`(`^`(GAMMA(`/`(2, 3)), 2), `*`(hypergeom([], [`/`(4, 3)], `+`(`*`(`/`(1, 9), `*`(`^`(x, 3)))))))))), `*`(Pi)...
[AiryAi(x) = `+`(`/`(`*`(`/`(1, 6), `*`(`+`(`-`(`/`(`*`(3, `*`(x, `*`(`^`(3, `/`(1, 6)), `*`(`^`(GAMMA(`/`(2, 3)), 2), `*`(hypergeom([], [`/`(4, 3)], `+`(`*`(`/`(1, 9), `*`(`^`(x, 3)))))))))), `*`(Pi)...
Plot_2d
 

Veamos cono se define la funci?n AiryBi(x): 

> FunctionAdvisor(definition,AiryBi(x));plot(AiryBi(x),x=-3..3);
 

 

[AiryBi(x) = `+`(`/`(`*`(`/`(1, 6), `*`(`+`(`/`(`*`(3, `*`(`^`(3, `/`(2, 3)), `*`(x, `*`(`^`(GAMMA(`/`(2, 3)), 2), `*`(hypergeom([], [`/`(4, 3)], `+`(`*`(`/`(1, 9), `*`(`^`(x, 3)))))))))), `*`(Pi)), `...
[AiryBi(x) = `+`(`/`(`*`(`/`(1, 6), `*`(`+`(`/`(`*`(3, `*`(`^`(3, `/`(2, 3)), `*`(x, `*`(`^`(GAMMA(`/`(2, 3)), 2), `*`(hypergeom([], [`/`(4, 3)], `+`(`*`(`/`(1, 9), `*`(`^`(x, 3)))))))))), `*`(Pi)), `...
Plot_2d
 

Ambas dependen de la Funci?n Hipergeom?trica Generalizada y de la funcion GAMMA. 

Definicion de la funci?n hipergeom?trica: 

> FunctionAdvisor(definition,hypergeom);
 

[hypergeom([a, b], [c], z) = Sum(`/`(`*`(pochhammer(a, _k1), `*`(pochhammer(b, _k1), `*`(`^`(z, _k1)))), `*`(factorial(_k1), `*`(pochhammer(c, _k1)))), _k1 = 0 .. infinity), `or`(`or`(`or`(And(`::`(a,...
[hypergeom([a, b], [c], z) = Sum(`/`(`*`(pochhammer(a, _k1), `*`(pochhammer(b, _k1), `*`(`^`(z, _k1)))), `*`(factorial(_k1), `*`(pochhammer(c, _k1)))), _k1 = 0 .. infinity), `or`(`or`(`or`(And(`::`(a,...
[hypergeom([a, b], [c], z) = Sum(`/`(`*`(pochhammer(a, _k1), `*`(pochhammer(b, _k1), `*`(`^`(z, _k1)))), `*`(factorial(_k1), `*`(pochhammer(c, _k1)))), _k1 = 0 .. infinity), `or`(`or`(`or`(And(`::`(a,...
 

Donde el simbolo pochhammer tiene el siguiente significado: 

> FunctionAdvisor(definition,pochhammer);
 

[pochhammer(z, n) = `/`(`*`(GAMMA(`+`(z, n))), `*`(GAMMA(z))), And(`::`(z, Not(nonposint)), `::`(`+`(z, n), Not(nonposint)))]
 

La funci?n GAMMA 

> FunctionAdvisor(definition,GAMMA(x));plot(GAMMA(x),x=-3..1);
 

 

[GAMMA(x) = Int(`/`(`*`(`^`(_k1, `+`(x, `-`(1)))), `*`(exp(_k1))), _k1 = 0 .. infinity), And(`<`(0, Re(x)))]
Plot_2d
 

Las funciones Lommel 

Las funci?nes Lommel se definen en funci?n de la serie hipergeometrica. Estas funciones son soluci?nes 

de la ecuacion diferencial lineal no homog?nea de segundo orden.
Veamos como se definen: 

> FunctionAdvisor(definition,LommelS1);FunctionAdvisor(definition,LommelS2);
 

 

[LommelS1(a, b, z) = `+`(`-`(`/`(`*`(`^`(z, `+`(a, 1)), `*`(hypergeom([1], [`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(b))), `*`(`/`(1, 2), `*`(a))), `+`(`/`(3, 2), `*`(`/`(1, 2), `*`(b)), `*`(`/`(1, 2), `...
[LommelS1(a, b, z) = `+`(`-`(`/`(`*`(`^`(z, `+`(a, 1)), `*`(hypergeom([1], [`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(b))), `*`(`/`(1, 2), `*`(a))), `+`(`/`(3, 2), `*`(`/`(1, 2), `*`(b)), `*`(`/`(1, 2), `...
[LommelS1(a, b, z) = `+`(`-`(`/`(`*`(`^`(z, `+`(a, 1)), `*`(hypergeom([1], [`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(b))), `*`(`/`(1, 2), `*`(a))), `+`(`/`(3, 2), `*`(`/`(1, 2), `*`(b)), `*`(`/`(1, 2), `...
[LommelS2(a, b, z) = `+`(LommelS1(a, b, z), `*`(`^`(2, `+`(a, `-`(1))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `-`(`*`(`/`(1, 2), `*`(b))), `/`(1, 2))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `*`(`/`(1,...
[LommelS2(a, b, z) = `+`(LommelS1(a, b, z), `*`(`^`(2, `+`(a, `-`(1))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `-`(`*`(`/`(1, 2), `*`(b))), `/`(1, 2))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `*`(`/`(1,...
[LommelS2(a, b, z) = `+`(LommelS1(a, b, z), `*`(`^`(2, `+`(a, `-`(1))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `-`(`*`(`/`(1, 2), `*`(b))), `/`(1, 2))), `*`(GAMMA(`+`(`*`(`/`(1, 2), `*`(a)), `*`(`/`(1,...
 

> plot(LommelS1(1,1,x),x=-2..2);plot(LommelS2(1,1,x),x=0..0.5);
 

 

Plot_2d
Plot_2d
 

 

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