#### 2.315   ODE No. 315

$\left (2 x y(x)^3-x^4\right ) y'(x)+2 x^3 y(x)-y(x)^4=0$ Mathematica : cpu = 0.302373 (sec), leaf count = 368

$\left \{\left \{y(x)\to \frac {\sqrt [3]{\frac {2}{3}} e^{c_1} x}{\sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}+\frac {\sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}{\sqrt [3]{2} 3^{2/3}}\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) e^{c_1} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}{2 \sqrt [3]{2} 3^{2/3}}\right \},\left \{y(x)\to -\frac {\left (1-i \sqrt {3}\right ) e^{c_1} x}{2^{2/3} \sqrt [3]{3} \sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^3+\sqrt {3} \sqrt {27 x^6-4 e^{3 c_1} x^3}}}{2 \sqrt [3]{2} 3^{2/3}}\right \}\right \}$ Maple : cpu = 0.072 (sec), leaf count = 376

$\left \{ y \left ( x \right ) ={\frac {\sqrt [3]{12}}{6\,{\it \_C1}} \left ( x\sqrt [3]{12}{\it \_C1}+ \left ( x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2}}}}},y \left ( x \right ) =-{\frac {\sqrt [3]{12}}{12\,{\it \_C1}} \left ( i\sqrt [3]{12}\sqrt {3}{\it \_C1}\,x-i \left ( x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}}\sqrt {3}+x\sqrt [3]{12}{\it \_C1}+ \left ( x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2}}}}},y \left ( x \right ) =-{\frac {\sqrt [3]{12}}{12\,{\it \_C1}} \left ( \left ( -ix{\it \_C1}\,\sqrt [3]{12}+i \left ( x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+x\sqrt [3]{12}{\it \_C1}+ \left ( x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{x \left ( -9\,{\it \_C1}\,{x}^{2}+\sqrt {3}\sqrt {{\frac {27\,{{\it \_C1}}^{3}{x}^{4}-4\,x}{{\it \_C1}}}} \right ) {{\it \_C1}}^{2}}}}} \right \}$