4.3.34 \(y'(x)=e^x \left (a+b e^{-y(x)}\right )\)

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

\[ \left \{ {{\rm e}^{x}}+{\frac {\ln \left ({{\rm e}^{-y \relax (x ) }} \right ) }{a}}-{\frac {\ln \left (a+b{{\rm e}^{-y \relax (x ) }} \right ) }{a}}+{\it \_C1}=0 \right \} \] 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