ODE
\[ x^2+x (2 y(x)+x) y''(x)+2 x y'(x)^2+4 (y(x)+x) y'(x)+2 y(x)=0 \] ODE Classification
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0835132 (sec), leaf count = 104
\[\left \{\left \{y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (12 c_2 x+12 c_1-x^4+3 x^3\right )}-3 x\right )\right \},\left \{y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (12 c_2 x+12 c_1-x^4+3 x^3\right )}-3 x\right )\right \}\right \}\]
Maple ✓
cpu = 0.044 (sec), leaf count = 28
\[ \left \{ -{\frac {{x}^{4}}{12}}-{x}^{2}y \left ( x \right ) -x \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}\,x+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x^2 + 2*y[x] + 4*(x + y[x])*y'[x] + 2*x*y'[x]^2 + x*(x + 2*y[x])*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-3*x - Sqrt[3]*Sqrt[x^(-2)]*Sqrt[x*(3*x^3 - x^4 + 12*C[1] + 12*x*C[2])])/6}, {y[x] -> (-3*x + Sqrt[3]*Sqrt[x^(-2)]*Sqrt[x*(3*x^3 - x^4 + 12*C[1] + 12*x*C[2])])/6}}
Maple raw input
dsolve(x*(x+2*y(x))*diff(diff(y(x),x),x)+2*x*diff(y(x),x)^2+4*(x+y(x))*diff(y(x),x)+x^2+2*y(x) = 0, y(x),'implicit')
Maple raw output
-1/12*x^4-x^2*y(x)-x*y(x)^2-_C1*x+_C2 = 0