#### 2.647   ODE No. 647

$y'(x)=\frac {x \left (a y(x)^2+b x^2\right )^2}{a^{5/2} y(x)}$ Mathematica : cpu = 0.361197 (sec), leaf count = 115

$\left \{\left \{y(x)\to -\sqrt {-\frac {b x^2}{a}+\frac {\sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{\sqrt [4]{a}}}\right \},\left \{y(x)\to \sqrt {-\frac {b x^2}{a}+\frac {\sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{\sqrt [4]{a}}}\right \}\right \}$ Maple : cpu = 0.492 (sec), leaf count = 460

$\left \{ y \left ( x \right ) ={\frac {1}{a}\sqrt {-a \left ( {\it \_C1}\,{{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) \left ( \left ( b{x}^{2}-{a}^{{\frac {3}{2}}}\sqrt {-{b{a}^{-{\frac {3}{2}}}}} \right ) {{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{\it \_C1}\, \left ( {a}^{{\frac {3}{2}}}\sqrt {-{b{a}^{-{\frac {3}{2}}}}}+b{x}^{2} \right ) {{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) } \left ( {\it \_C1}\,{{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) ^{-1}},y \left ( x \right ) =-{\frac {1}{a}\sqrt {-a \left ( {\it \_C1}\,{{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) \left ( \left ( b{x}^{2}-{a}^{{\frac {3}{2}}}\sqrt {-{b{a}^{-{\frac {3}{2}}}}} \right ) {{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{\it \_C1}\, \left ( {a}^{{\frac {3}{2}}}\sqrt {-{b{a}^{-{\frac {3}{2}}}}}+b{x}^{2} \right ) {{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) } \left ( {\it \_C1}\,{{\rm e}^{{\frac {{x}^{2}}{2} \left ( 2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}}+{{\rm e}^{{\frac {{x}^{2}}{2} \left ( -2\,{a}^{3/2}\sqrt {-{\frac {b}{{a}^{3/2}}}}+b{x}^{2} \right ) {a}^{-{\frac {3}{2}}}}}} \right ) ^{-1}} \right \}$