ODE
\[ y'(x)=e^x \left (a+b e^{-y(x)}\right ) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.0220513 (sec), leaf count = 24
\[\left \{\left \{y(x)\to \log \left (\frac {e^{a \left (c_1+e^x\right )}-b}{a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 32
\[ \left \{ {{\rm e}^{x}}+{\frac {\ln \left ( {{\rm e}^{-y \left ( x \right ) }} \right ) }{a}}-{\frac {\ln \left ( a+b{{\rm e}^{-y \left ( x \right ) }} \right ) }{a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y'[x] == E^x*(a + b/E^y[x]),y[x],x]
Mathematica raw output
{{y[x] -> Log[(-b + E^(a*(E^x + C[1])))/a]}}
Maple raw input
dsolve(diff(y(x),x) = exp(x)*(a+b*exp(-y(x))), y(x),'implicit')
Maple raw output
exp(x)+1/a*ln(exp(-y(x)))-1/a*ln(a+b*exp(-y(x)))+_C1 = 0