Titchmarsh ODEs - Maple Help

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Titchmarsh ODEs



The general form of the Titchmarsh ODE is given by:

Titchmarsh_ode := diff(y(x),x,x)+(lambda-x^(2*n))*y(x)=0;



where n is an integer. See Hille, "Lectures on Ordinary Differential Equations", p. 617.


All linear second order homogeneous ODEs can be transformed into first order ODEs of Riccati type by giving the symmetry [0,y] to dsolve (all linear homogeneous ODEs have this symmetry) or by calling convert (see convert,ODEs).






Reduction to Riccati by giving the symmetry to dsolve


ans:=yx=ⅇ∫_b_aⅆ_a+_C1 &where ⅆⅆ_a_b_a=_b_a2+_a2nλ,_a=x,_b_a=ⅆⅆxyxyx,x=_a,yx=ⅇ∫_b_aⅆ_a+_C1


The reduced ODE above is of Riccati type:







Converting this ODE into a first order ODE of Riccati type




In the answer returned by convert, there are the Riccati ODE and the transformation of variables used. Changes of variables in ODEs can be performed using ?PDEtools[dchange]. For example, using the transformation of variables above, we can recover the result returned by convert.

See Also

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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