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 = 0.217609 (sec), leaf count = 104
\[\left \{\left \{y(x)\to \frac {1}{6} \left (-3 x-\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right )\right \},\left \{y(x)\to \frac {1}{6} \left (-3 x+\sqrt {3} \sqrt {\frac {1}{x^2}} \sqrt {x \left (-x^4+3 x^3+12 c_2 x+12 c_1\right )}\right )\right \}\right \}\]
Maple ✓
cpu = 0.206 (sec), leaf count = 75
\[\left [y \left (x \right ) = \frac {-3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-36 x^{2} \textit {\_C1} +36 \textit {\_C2} x}}{6 x}, y \left (x \right ) = -\frac {3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-36 x^{2} \textit {\_C1} +36 \textit {\_C2} x}}{6 x}\right ]\] 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))
Maple raw output
[y(x) = 1/6*(-3*x^2+(-3*x^5+9*x^4-36*_C1*x^2+36*_C2*x)^(1/2))/x, y(x) = -1/6*(3*x^2+(-3*x^5+9*x^4-36*_C1*x^2+36*_C2*x)^(1/2))/x]