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Titchmarsh ODEs
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Description
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The general form of the Titchmarsh ODE is given by:
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Titchmarsh_ode := diff(y(x),x,x)+(lambda-x^(2*n))*y(x)=0;
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where n is an integer. See Hille, "Lectures on Ordinary Differential Equations", p. 617.
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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).
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Examples
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![[_Titchmarsh]](/support/helpjp/helpview.aspx?si=7245/file04362/math40.png)
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Reduction to Riccati by giving the symmetry to dsolve
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![ans := dsolve(Titchmarsh_ode, HINT = [0, y])](/support/helpjp/helpview.aspx?si=7245/file04362/math49.png)
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![ans := y(x) = `&where`(exp(Int(_b(_a), _a)+_C1), [{diff(_b(_a), _a) = -_b(_a)^2-lambda+_a^(2*n)}, {_a = x, _b(_a) = (diff(y(x), x))/y(x)}, {x = _a, y(x) = exp(Int(_b(_a), _a)+_C1)}])](/support/helpjp/helpview.aspx?si=7245/file04362/math52.png)
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The reduced ODE above is of Riccati type:
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![reduced_ode := op([2, 2, 1, 1], ans)](/support/helpjp/helpview.aspx?si=7245/file04362/math58.png)
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![[_Riccati]](/support/helpjp/helpview.aspx?si=7245/file04362/math68.png)
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Converting this ODE into a first order ODE of Riccati type
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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.
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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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