2.401   ODE No. 401

$3 y'(x)^2-2 x y'(x)+y(x)=0$ Mathematica : cpu = 0.308421 (sec), leaf count = 1093

$\left \{\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+3 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-24 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+243 e^{12 c_1}+48 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [-16 e^{6 c_1} x^6+243 \text {\#1}^4 x^4+144 e^{6 c_1} \text {\#1} x^4-1944 \text {\#1}^5 x^2-378 e^{6 c_1} \text {\#1}^2 x^2+3 e^{12 c_1}+3888 \text {\#1}^6+216 e^{6 c_1} \text {\#1}^3\& ,6\right ]\right \}\right \}$ Maple : cpu = 0.112 (sec), leaf count = 580

$\left \{ y \left ( x \right ) =-{\frac {1}{48} \left ( i \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}- \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+2\,x\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}-{x}^{2} \right ) \left ( i \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}- \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-6\,x\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}-{x}^{2} \right ) \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{-{\frac {2}{3}}}},y \left ( x \right ) =-{\frac {1}{48} \left ( i \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}+ \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-2\,x\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}+{x}^{2} \right ) \left ( i \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}\sqrt {3}-i\sqrt {3}{x}^{2}+ \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+6\,x\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}+{x}^{2} \right ) \left ( -54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}} \right ) ^{-{\frac {2}{3}}}},y \left ( x \right ) =-{\frac {1}{12} \left ( {{x}^{2}{\frac {1}{\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}}}}+x+\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}} \right ) ^{2}}+{\frac {x}{3} \left ( {{x}^{2}{\frac {1}{\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}}}}}+x+\sqrt [3]{-54\,{\it \_C1}+{x}^{3}+6\,\sqrt {-3\,{\it \_C1}\,{x}^{3}+81\,{{\it \_C1}}^{2}}} \right ) } \right \}$