ODE
\[ a e^{y(x)}+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0792714 (sec), leaf count = 35
\[\left \{\left \{y(x)\to \log \left (\frac {c_1 \text {sech}^2\left (\frac {1}{2} \sqrt {c_1 \left (c_2+x\right ){}^2}\right )}{2 a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.097 (sec), leaf count = 55
\[ \left \{ -2\,{\it \_C1}\,{\it Artanh} \left ( \sqrt {-2\,a{{\rm e}^{y \left ( x \right ) }}+{{\it \_C1}}^{-2}}{\it \_C1} \right ) -x-{\it \_C2}=0,2\,{\it \_C1}\,{\it Artanh} \left ( \sqrt {-2\,a{{\rm e}^{y \left ( x \right ) }}+{{\it \_C1}}^{-2}}{\it \_C1} \right ) -x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[a*E^y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Log[(C[1]*Sech[Sqrt[C[1]*(x + C[2])^2]/2]^2)/(2*a)]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*exp(y(x)) = 0, y(x),'implicit')
Maple raw output
-2*_C1*arctanh((-2*a*exp(y(x))+1/_C1^2)^(1/2)*_C1)-x-_C2 = 0, 2*_C1*arctanh((-2*a*exp(y(x))+1/_C1^2)^(1/2)*_C1)-x-_C2 = 0