ODE
\[ a x e^{y(x)}+x y''(x)+y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.8465 (sec), leaf count = 190
\[\left \{\left \{y(x)\to \log \left (\frac {4 e^{c_2 \sqrt {4+2 a c_1}} (2+a c_1){}^2 x^{-2+\sqrt {4+2 a c_1}}}{\left (a^2 c_1 e^{c_2 \sqrt {4+2 a c_1}}+x^{\sqrt {4+2 a c_1}}+2 a e^{c_2 \sqrt {4+2 a c_1}}\right ){}^2}\right )\right \},\left \{y(x)\to \log \left (\frac {4 e^{c_2 \sqrt {4+2 a c_1}} (2+a c_1){}^2 x^{-2+\sqrt {4+2 a c_1}}}{\left (a (2+a c_1) x^{\sqrt {4+2 a c_1}}+e^{c_2 \sqrt {4+2 a c_1}}\right ){}^2}\right )\right \}\right \}\]
Maple ✓
cpu = 2.205 (sec), leaf count = 62
\[\left [y \left (x \right ) = \ln \left (-\frac {\left (\tan ^{2}\left (-\frac {\ln \left (x \right ) \sqrt {\textit {\_C1} -4}}{2}+\frac {\textit {\_C2} \sqrt {\textit {\_C1} -4}}{2}\right )\right ) \textit {\_C1} -4 \left (\tan ^{2}\left (-\frac {\ln \left (x \right ) \sqrt {\textit {\_C1} -4}}{2}+\frac {\textit {\_C2} \sqrt {\textit {\_C1} -4}}{2}\right )\right )+\textit {\_C1} -4}{2 a \,x^{2}}\right )\right ]\] Mathematica raw input
DSolve[a*E^y[x]*x + y'[x] + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Log[(4*E^(Sqrt[4 + 2*a*C[1]]*C[2])*x^(-2 + Sqrt[4 + 2*a*C[1]])*(2 + a*C[1])^2)/(2*a*E^(Sqrt[4 + 2*a*C[1]]*C[2]) + x^Sqrt[4 + 2*a*C[1]] + a^2*E^(Sqrt[4 + 2*a*C[1]]*C[2])*C[1])^2]}, {y[x] -> Log[(4*E^(Sqrt[4 + 2*a*C[1]]*C[2])*x^(-2 + Sqrt[4 + 2*a*C[1]])*(2 + a*C[1])^2)/(E^(Sqrt[4 + 2*a*C[1]]*C[2]) + a*x^Sqrt[4 + 2*a*C[1]]*(2 + a*C[1]))^2]}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+diff(y(x),x)+a*x*exp(y(x)) = 0, y(x))
Maple raw output
[y(x) = ln(-1/2*(tan(-1/2*ln(x)*(_C1-4)^(1/2)+1/2*_C2*(_C1-4)^(1/2))^2*_C1-4*tan(-1/2*ln(x)*(_C1-4)^(1/2)+1/2*_C2*(_C1-4)^(1/2))^2+_C1-4)/a/x^2)]