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Jacobi ODEs
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Description
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The general form of the Jacobi ODE is given by the following:
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Jacobi_ode := diff(y(x),x,x)*x*(1-x) = (g-(a+1)*x)*diff(y(x),x)+n*(a+n)*y(x);
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where n is an integer. See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1480.
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Examples
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The solution to this type of ODE can be expressed in terms of the hypergeometric function; see hypergeom.
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![[odeadvisor]](/support/helpjp/helpview.aspx?si=7238/file04355/math33.png)
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![[_Jacobi]](/support/helpjp/helpview.aspx?si=7238/file04355/math40.png)
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![y(x) = _C1*hypergeom([-1-(1/2)*a-(1/2)*(a^2+(-4*n+4)*a+4-4*n^2)^(1/2), -1-(1/2)*a+(1/2)*(a^2+(-4*n+4)*a+4-4*n^2)^(1/2)], [-g], x)+_C2*x^(1+g)*hypergeom([-(1/2)*a-(1/2)*(a^2+(-4*n+4)*a+4-4*n^2)^(1/2)+g, -(1/2)*a+(1/2)*(a^2+(-4*n+4)*a+4-4*n^2)^(1/2)+g], [2+g], x)](/support/helpjp/helpview.aspx?si=7238/file04355/math47.png)
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See Also
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DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.
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