I came across an ordinary differential equation of the form:
$y''+yy' + Ay + A = 0,$
Where $y=y(t)$, $A$ is some constant, positive or negative. So far the equation is solvable for $A=0$. Could somebody help in finding a general solution? The equation does not depend explicitly on $t$, and yet it is challenging to solve it. I'm interested mostly in an analytical expression including special functions, series, etc., as long as there is such a solution. As a last resort, I'd go with polynomials or define special ones if needed.
Thank you
Let $y'(t)=p(y)$ and substitute in the ode,. Then $y''=p p'$ and ode becomes first order chini ode $$ P'(y)=-y-A(1+y)P^{-1} $$ Chini ode are not all solvable. I do not now have access to my PC to try solving this. But if the above can be solved for $P$, then you can now solve the original ode as first order ode using $y'=P$