ODE
\[ a y(x) y'(x)+x y(x) y''(x)+2 x y'(x)^2=0 \] ODE Classification
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.405506 (sec), leaf count = 29
\[\left \{\left \{y(x)\to c_2 \sqrt [3]{3 x^{1-a}-a c_1+c_1}\right \}\right \}\]
Maple ✓
cpu = 0.291 (sec), leaf count = 234
\[\left [y \left (x \right ) = \frac {x^{-a} \left (\left (3 \textit {\_C2} \,x^{a} a -3 \textit {\_C2} \,x^{a}-3 \textit {\_C1} x \right ) x^{2 a} \left (a -1\right )^{2}\right )^{\frac {1}{3}}}{a -1}, y \left (x \right ) = -\frac {x^{-a} \left (\left (3 \textit {\_C2} \,x^{a} a -3 \textit {\_C2} \,x^{a}-3 \textit {\_C1} x \right ) x^{2 a} \left (a -1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (a -1\right )}-\frac {i \sqrt {3}\, x^{-a} \left (\left (3 \textit {\_C2} \,x^{a} a -3 \textit {\_C2} \,x^{a}-3 \textit {\_C1} x \right ) x^{2 a} \left (a -1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (a -1\right )}, y \left (x \right ) = -\frac {x^{-a} \left (\left (3 \textit {\_C2} \,x^{a} a -3 \textit {\_C2} \,x^{a}-3 \textit {\_C1} x \right ) x^{2 a} \left (a -1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (a -1\right )}+\frac {i \sqrt {3}\, x^{-a} \left (\left (3 \textit {\_C2} \,x^{a} a -3 \textit {\_C2} \,x^{a}-3 \textit {\_C1} x \right ) x^{2 a} \left (a -1\right )^{2}\right )^{\frac {1}{3}}}{-2+2 a}\right ]\] Mathematica raw input
DSolve[a*y[x]*y'[x] + 2*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (3*x^(1 - a) + C[1] - a*C[1])^(1/3)*C[2]}}
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)^2+a*y(x)*diff(y(x),x) = 0, y(x))
Maple raw output
[y(x) = 1/(x^a)/(a-1)*((3*_C2*x^a*a-3*_C2*x^a-3*_C1*x)*(x^a)^2*(a-1)^2)^(1/3), y(x) = -1/2/(x^a)/(a-1)*((3*_C2*x^a*a-3*_C2*x^a-3*_C1*x)*(x^a)^2*(a-1)^2)^(1/3)-1/2*I*3^(1/2)/(x^a)/(a-1)*((3*_C2*x^a*a-3*_C2*x^a-3*_C1*x)*(x^a)^2*(a-1)^2)^(1/3), y(x) = -1/2/(x^a)/(a-1)*((3*_C2*x^a*a-3*_C2*x^a-3*_C1*x)*(x^a)^2*(a-1)^2)^(1/3)+1/2*I*3^(1/2)/(x^a)/(a-1)*((3*_C2*x^a*a-3*_C2*x^a-3*_C1*x)*(x^a)^2*(a-1)^2)^(1/3)]