Solve second order ode using lie symmetry condition and explain steps clearly in detail. ( ie canonical variable,lie point symmetries,r , s etc ) yy'' - y'^2  = 3y^2 lny( Full calculation and answer ).   I have the infinitesimal generator X = d/dx. Can I solve it, find general solution, by using lie symmetry condition, canonical coordinates(r and s), canonical variables.   I have to follow these steps to solve this question: 1) find the lie symmetries, 2) find canonical coordinates, 3) substituting canonical variables, 4) solving ODE in new coordinates, 5) Returning to original coordinates- general solution.   Can u help me follow these steps in order to answer the question above please?’—> that was the original question. I got y'y'' + yy''' - 2y'y'' = 3y + 6ylny from subbing in X= d/dx, differentiating. I then simplified this equation to: yy''' - y'y'' = 3y + 6ylny, simplifying the original equation. I then introduce canonical coordinates r = y’ and s = y” and substitute that into into yy''' - y'y'' = 3y + 6ylny; I get: yrs-rs= 3y + 6ylny. Is everything correct up to this point? Can u show me the calculations for the next few steps as well with the explanations please?

Solve second order ode using lie symmetry condition and explain steps clearly in detail. ( ie canonical variable,lie point symmetries,r , s etc )

yy'' - y'^2  = 3y^2 lny( Full calculation and answer ).

I have the infinitesimal generator X = d/dx. Can I solve it, find general solution, by using lie symmetry condition, canonical coordinates(r and s), canonical variables.

I have to follow these steps to solve this question: 1) find the lie symmetries, 2) find canonical coordinates, 3) substituting canonical variables, 4) solving ODE in new coordinates, 5) Returning to original coordinates- general solution.

Can u help me follow these steps in order to answer the question above please?’—> that was the original question. I got y'y'' + yy''' - 2y'y'' = 3y + 6ylny from subbing in X= d/dx, differentiating. I then simplified this equation to: yy''' - y'y'' = 3y + 6ylny, simplifying the original equation. I then introduce canonical coordinates r = y’ and s = y” and substitute that into into yy''' - y'y'' = 3y + 6ylny; I get: yrs-rs= 3y + 6ylny. Is everything correct up to this point? Can u show me the calculations for the next few steps as well with the explanations please?