Verify that the 3rd ODE
$\displaystyle \mu^2\mu'''=\nu (2\mu \mu''-\mu'^2)$
admit two infinitesimal generator
$\displaystyle X_1=\frac{\partial}{\partial x}, X_2=x\frac{\partial}{\partial x}+\mu\frac{\partial}{\partial \mu}.$
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