3.270   ODE No. 270

\[ \boxed { \left ( \left ( y \left ( x \right ) \right ) ^{2}-x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -y \left ( x \right ) +{x}^{2}=0} \]

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

Mathematica: cpu = 0.014502 (sec), leaf count = 327 \[ \left \{\left \{y(x)\to -\frac {3 \sqrt [3]{2} x}{\sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}-\frac {\sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}{3 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {3 \left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}{6 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {3 \left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (81 c_1+27 x^3\right ){}^2-2916 x^3}+81 c_1+27 x^3}}{6 \sqrt [3]{2}}\right \}\right \} \]

Maple: cpu = 0.031 (sec), leaf count = 402 \[ \left \{ y \left ( x \right ) ={\frac {1}{2}\sqrt [3]{-4\,{x}^{3}-12\,{ \it \_C1}+4\,\sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}+2\,{\frac {x}{\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4\, \sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}} ,y \left ( x \right ) =-{\frac {1}{4}\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1 }+4\,\sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2 }}}}-{x{\frac {1}{\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4\,\sqrt {{x}^{ 6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}}}-{\frac {i }{2}}\sqrt {3} \left ( {\frac {1}{2}\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1 }+4\,\sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2 }}}}-2\,{\frac {x}{\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4\,\sqrt {{x}^ {6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}} \right ) ,y \left ( x \right ) =-{\frac {1}{4}\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+ 4\,\sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}} }}-{x{\frac {1}{\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4\,\sqrt {{x}^{6} +6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}}}+{\frac {i}{2 }}\sqrt {3} \left ( {\frac {1}{2}\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4 \,\sqrt {{x}^{6}+6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}} }-2\,{\frac {x}{\sqrt [3]{-4\,{x}^{3}-12\,{\it \_C1}+4\,\sqrt {{x}^{6} +6\,{x}^{3}{\it \_C1}-4\,{x}^{3}+9\,{{\it \_C1}}^{2}}}}} \right ) \right \} \]