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 = 1.26248 (sec), leaf count = 190
\[\left \{\left \{y(x)\to \log \left (\frac {4 e^{c_2 \sqrt {2 a c_1+4}} \left (a c_1+2\right ){}^2 x^{\sqrt {2 a c_1+4}-2}}{\left (a^2 c_1 e^{c_2 \sqrt {2 a c_1+4}}+x^{\sqrt {2 a c_1+4}}+2 a e^{c_2 \sqrt {2 a c_1+4}}\right ){}^2}\right )\right \},\left \{y(x)\to \log \left (\frac {4 e^{c_2 \sqrt {2 a c_1+4}} \left (a c_1+2\right ){}^2 x^{\sqrt {2 a c_1+4}-2}}{\left (a \left (a c_1+2\right ) x^{\sqrt {2 a c_1+4}}+e^{c_2 \sqrt {2 a c_1+4}}\right ){}^2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.458 (sec), leaf count = 77
\[ \left \{ -2\,{\frac {1}{\sqrt {{\it \_C1}-4}}\arctan \left ( {\frac {\sqrt {4-2\,{{\rm e}^{y \left ( x \right ) }}a{x}^{2}-{\it \_C1}}}{\sqrt {{\it \_C1}-4}}} \right ) }+\ln \left ( x \right ) -{\it \_C2}=0,2\,{\frac {1}{\sqrt {{\it \_C1}-4}}\arctan \left ( {\frac {\sqrt {4-2\,{{\rm e}^{y \left ( x \right ) }}a{x}^{2}-{\it \_C1}}}{\sqrt {{\it \_C1}-4}}} \right ) }+\ln \left ( x \right ) -{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
2/(_C1-4)^(1/2)*arctan((4-2*exp(y(x))*a*x^2-_C1)^(1/2)/(_C1-4)^(1/2))+ln(x)-_C2 = 0, -2/(_C1-4)^(1/2)*arctan((4-2*exp(y(x))*a*x^2-_C1)^(1/2)/(_C1-4)^(1/2))+ln(x)-_C2 = 0