The general form of the Titchmarsh ODE is given by:

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

$\mathrm{Titchmarsh\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\left(\mathrm{\lambda }-x^{2n}\right)y\left(x\right)=0$

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).

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)\:$

$\mathrm{odeadvisor}\left(\mathrm{Titchmarsh\_ode}\right)$

$\left[\mathrm{\_Titchmarsh}\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Titchmarsh\_ode}\,\mathrm{HINT}=\left[0\,y\right]\right)$

$\mathrm{ans}≔y\left(x\right)=\left({\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right)\mathbf{where}\left[\left\{\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=-{\mathrm{\_b}\left(\mathrm{\_a}\right)}^2+{\mathrm{\_a}}^{2n}-\mathrm{\lambda }\right\}\,\left\{\mathrm{\_a}=x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{y\left(x\right)}\right\}\,\left\{x=\mathrm{\_a}\,y\left(x\right)={\ⅇ}^{\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}}\right\}\right]$

$\mathrm{reduced\_ode}≔\mathrm{op}\left(\left[2\,2\,1\,1\right]\,\mathrm{ans}\right)$

$\mathrm{reduced\_ode}≔\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)=-{\mathrm{\_b}\left(\mathrm{\_a}\right)}^2+{\mathrm{\_a}}^{2n}-\mathrm{\lambda }$

$\mathrm{odeadvisor}\left(\mathrm{reduced\_ode}\right)$

$\left[\mathrm{\_Riccati}\right]$

$\mathrm{Riccati\_ode\_TR}≔\mathrm{convert}\left(\mathrm{Titchmarsh\_ode}\,\mathrm{Riccati}\right)$

$\mathrm{Riccati\_ode\_TR}≔\frac{\ⅆ}{\ⅆx}\mathrm{\_a}\left(x\right)=\mathrm{\_F1}\left(x\right){\mathrm{\_a}\left(x\right)}^2-\frac{\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\mathrm{\_a}\left(x\right)}{\mathrm{\_F1}\left(x\right)}+\frac{\mathrm{\lambda }-x^{2n}}{\mathrm{\_F1}\left(x\right)},\left\{y\left(x\right)={\ⅇ}^{-\left(\int \mathrm{\_a}\left(x\right)\mathrm{\_F1}\left(x\right)\ⅆx\right)}\mathrm{c\_\_1}\right\}$