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);

are given by:

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

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

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

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

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.