Solution of a riccati equation. We're given a self-adjoint equation as follows : $$ \dfrac{d}{dt}[t \frac{dx}{dt}]+(1-t)x = 0$$ We first convert this into a riccati equation , and hence we get : $$ \dfrac{du}{dt} + (\dfrac{1}{t})u^{2} +(1-t)=0$$ Now we want the solution of this riccati equation of the form $ct^{n}$. How to obtain this ? 
All I know about riccati equation is that the substitution $u = f + \frac{1}{v}$ would convert this system into a known system . But for this we should have a solution $f$ to the system above, and no such solution is given in the question. Can anyone help ?

As you said you must have a solution to get the other one. by a simple guess you can make the guess that solution is $u=-t$. and then find the other root. the first root must be given in problem or else you must guess it.
