#### 2.273   ODE No. 273

$\left (a+x^2+y(x)^2\right ) y'(x)+2 x y(x)=0$ Mathematica : cpu = 0.164338 (sec), leaf count = 297

$\left \{\left \{y(x)\to \frac {\sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}{3 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (a+x^2\right )}{\sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}\right \},\left \{y(x)\to \frac {3 \left (1+i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}{6 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {3 \left (1-i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+6561 c_1{}^2}+81 c_1}}{6 \sqrt [3]{2}}\right \}\right \}$ Maple : cpu = 0.023 (sec), leaf count = 401

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