Setup: We have (all) solutions $y_k(x)$ of an inhomogeneous, (possibly non-linear) first-order ODE
$P(x)(y)=0$
where the coefficients of $P(x)(y)$ are monomials in $x$.
Question: Are we now able to find solutions of $P(x+1)(y)=0$?
The Problem where this question occured is the non-linear ODE
$(x+1)^5(y')^2+3(x+1)^4y'y+\frac94(x+1)^3y^2+\frac{\pi^2}4*(1-(x+1)^3)=0$