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;
 ${\mathrm{Titchmarsh_ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}\left({\mathrm{λ}}{-}{{x}}^{{2}{}{n}}\right){}{y}{}\left({x}\right){=}{0}$ (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

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right):$
 > $\mathrm{odeadvisor}\left(\mathrm{Titchmarsh_ode}\right)$
 $\left[{\mathrm{_Titchmarsh}}\right]$ (2)

Reduction to Riccati by giving the symmetry to dsolve

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Titchmarsh_ode},\mathrm{HINT}=\left[0,y\right]\right)$
 ${\mathrm{ans}}{:=}{y}{}\left({x}\right){=}\left({{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right){&where}\left[\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}{-}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}{+}{{\mathrm{_a}}}^{{2}{}{n}}{-}{\mathrm{λ}}\right\}{,}\left\{{\mathrm{_a}}{=}{x}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)}{{y}{}\left({x}\right)}\right\}{,}\left\{{x}{=}{\mathrm{_a}}{,}{y}{}\left({x}\right){=}{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right\}\right]$ (3)

The reduced ODE above is of Riccati type:

 > $\mathrm{reduced_ode}≔\mathrm{op}\left(\left[2,2,1,1\right],\mathrm{ans}\right)$
 ${\mathrm{reduced_ode}}{:=}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}{-}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}{+}{{\mathrm{_a}}}^{{2}{}{n}}{-}{\mathrm{λ}}$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{reduced_ode}\right)$
 $\left[{\mathrm{_Riccati}}\right]$ (5)

Converting this ODE into a first order ODE of Riccati type

 > $\mathrm{Riccati_ode_TR}≔\mathrm{convert}\left(\mathrm{Titchmarsh_ode},\mathrm{Riccati}\right)$
 ${\mathrm{Riccati_ode_TR}}{:=}\frac{{ⅆ}}{{ⅆ}{x}}{}{\mathrm{_a}}{}\left({x}\right){=}{\mathrm{_F1}}{}\left({x}\right){}{{\mathrm{_a}}{}\left({x}\right)}^{{2}}{-}\frac{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{\mathrm{_F1}}{}\left({x}\right)\right){}{\mathrm{_a}}{}\left({x}\right)}{{\mathrm{_F1}}{}\left({x}\right)}{+}\frac{{\mathrm{λ}}{-}{{x}}^{{2}{}{n}}}{{\mathrm{_F1}}{}\left({x}\right)}{,}\left\{{y}{}\left({x}\right){=}{{ⅇ}}^{{-}\left({∫}{\mathrm{_a}}{}\left({x}\right){}{\mathrm{_F1}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{}{\mathrm{_C1}}\right\}$ (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.