The general forms of ODEs having one of the following linear symmetries

[xi=a+b*x, eta=0], [xi=a+b*y, eta=0], [xi=0, eta=c+d*x], [xi=0, eta=c+d*y]:

where the infinitesimal symmetry generator is given by:

G := f -> xi*diff(f,x) + eta*diff(f,y);

$G≔f\→\mathrm{\ξ}\left(\frac{\partial }{\partial x}f\right)\+\mathrm{\η}\left(\frac{\partial }{\partial y}f\right)$

are given by:

ode[1] := DEtools[equinv]([xi=a+b*x, eta=0], y(x), 2);

${\mathrm{ode}}_1≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\mathrm{f\_\_1}\left(y\left(x\right)\,\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(bx+a\right)\right)}{{\left(bx+a\right)}^2}$

ode[2] := DEtools[equinv]([xi=a+b*y, eta=0], y(x), 2);

${\mathrm{ode}}_2≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\mathrm{f\_\_1}\left(y\left(x\right)\,-\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)bx-by\left(x\right)-a}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(by\left(x\right)+a\right)}\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^3}{{\left(by\left(x\right)+a\right)}^3}$

ode[3] := DEtools[equinv]([xi=0, eta=c+d*x], y(x), 2);

${\mathrm{ode}}_3≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\mathrm{f\_\_1}\left(x\,\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)dx+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)c-dy\left(x\right)}{xd+c}\right)$

ode[4] := DEtools[equinv]([xi=0, eta=c+d*y], y(x), 2);

${\mathrm{ode}}_4≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\mathrm{f\_\_1}\left(x\,\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{dy\left(x\right)+c}\right)dy\left(x\right)+\mathrm{f\_\_1}\left(x\,\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{dy\left(x\right)+c}\right)c$

Although the symmetries of these families of ODEs can be determined in a direct manner (using symgen), the simplicity of their pattern motivated us to have separate routines for recognizing them.

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

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[1\right]\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[2\right]\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[3\right]\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[4\right]\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{ode}\left[5\right]≔\mathrm{equinv}\left(\left[\left[0\,y\right]\,\left[x\,0\right]\right]\,y\left(x\right)\,2\right)$

${\mathrm{ode}}_5≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\mathrm{f\_\_1}\left(\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x}{y\left(x\right)}\right)y\left(x\right)}{x^2}$

$\mathrm{dsolve}\left(\mathrm{ode}\left[5\right]\right)$

$y\left(x\right)={\ⅇ}^{{\int }_{\phantom{\mathrm{`\; `}}}^{\ln \left(x\right)}\mathrm{RootOf}\left(-\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{-\mathrm{\_a}+{\mathrm{\_a}}^2-\mathrm{f\_\_1}\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}\right)-\mathrm{\_b}+\mathrm{c\_\_1}\right)\ⅆ\mathrm{\_b}+\mathrm{c\_\_2}}$