I wonder if the ODE

$y''+e^{u}=a$

can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions

$y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$. Are there explicit solutions for $a>0$?

I wonder if the ODE

$y''+e^{y}=a$

can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions

$y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$. Are there explicit solutions for $a>0$?