4.2.29 \(y'(x)=y(x)^2 \left (a e^x+y(x)\right )\)

ODE
\[ y'(x)=y(x)^2 \left (a e^x+y(x)\right ) \] ODE Classification

[_Abel]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 0.876621 (sec), leaf count = 73

\[\text {Solve}\left [-i a e^x=\frac {2 e^{-\frac {\left (a e^x y(x)+1\right )^2}{2 y(x)^2}}}{2 c_1-i \sqrt {2 \pi } \text {erf}\left (\frac {a e^x y(x)+1}{\sqrt {2} y(x)}\right )},y(x)\right ]\]

Maple
cpu = 0.062 (sec), leaf count = 50

\[ \left \{ {\it \_C1}+{\frac {1}{a{{\rm e}^{x}}}{{\rm e}^{-{\frac { \left (a{{\rm e}^{x}}+ \left (y \relax (x ) \right ) ^{-1} \right ) ^{2}}{2}}}}}+{\frac {\sqrt {2}\sqrt {\pi }}{2}{\it Erf} \left ({\frac { \left (a{{\rm e}^{x}}+ \left (y \relax (x ) \right ) ^{-1} \right ) \sqrt {2}}{2}} \right ) }=0 \right \} \] Mathematica raw input

DSolve[y'[x] == y[x]^2*(a*E^x + y[x]),y[x],x]

Mathematica raw output

Solve[(-I)*a*E^x == 2/(E^((1 + a*E^x*y[x])^2/(2*y[x]^2))*(2*C[1] - I*Sqrt[2*Pi]*
Erf[(1 + a*E^x*y[x])/(Sqrt[2]*y[x])])), y[x]]

Maple raw input

dsolve(diff(y(x),x) = (a*exp(x)+y(x))*y(x)^2, y(x),'implicit')

Maple raw output

_C1+exp(-1/2*(a*exp(x)+1/y(x))^2)/a/exp(x)+1/2*erf(1/2*(a*exp(x)+1/y(x))*2^(1/2)
)*2^(1/2)*Pi^(1/2) = 0