Titchmarsh ODEs - Maple Help

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Titchmarsh ODEs

Description

• 

The general form of the Titchmarsh ODE is given by:

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

Titchmarsh_ode:=ⅆ2ⅆx2yx+λx2nyx=0

(1)
  

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).

Examples

withDEtools,odeadvisor:

odeadvisorTitchmarsh_ode

_Titchmarsh

(2)

Reduction to Riccati by giving the symmetry to dsolve

ans:=dsolveTitchmarsh_ode,HINT=0,y

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

(3)

The reduced ODE above is of Riccati type:

reduced_ode:=op2,2,1,1,ans

reduced_ode:=ⅆⅆ_a_b_a=_b_a2+_a2nλ

(4)

odeadvisorreduced_ode

_Riccati

(5)

Converting this ODE into a first order ODE of Riccati type

Riccati_ode_TR:=convertTitchmarsh_ode,Riccati

Riccati_ode_TR:=ⅆⅆx_ax=_F1x_ax2ⅆⅆx_F1x_ax_F1x+λx2n_F1x,yx=ⅇ∫_ax_F1xⅆx_C1

(6)

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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