Verify that the 3rd ODE
admit two infinitesimal generator
And using these symmetry to simplify the above ode to a 1rst order ODE.
I could only verify the vector fields indeed are infinitesimal generators of the 3rd ODE. However, I do not know how to simplify the ODE. Would you help me?
This is Problem 6.3 in
A practical course in differential equations and mathematical modelling
by Nail H. Ibragimov,Nailʹ Khaĭrullovich Ibragimov