#### 2.402   ODE No. 402

$x^2+4 x y'(x)+3 y'(x)^2-y(x)=0$ Mathematica : cpu = 5.09782 (sec), leaf count = 1211

$\left \{\text {Solve}\left [\frac {1}{2} \left (\frac {\tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {1-9 y(x)}}\right )}{\sqrt {1-9 y(x)}}+\log \left (-3 x^2+2 x-12 y(x)+1\right )+\frac {9 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) y(x)}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}-\frac {9 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) y(x)}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}+\frac {\tanh ^{-1}\left (\frac {3 x-1}{2 \sqrt {1-9 y(x)}}\right )}{\sqrt {1-9 y(x)}}+\frac {2 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}-\frac {\tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}+\frac {2 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {1}{2} \left (\frac {\tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {1-9 y(x)}}\right )}{\sqrt {1-9 y(x)}}+\log \left (-3 x^2+2 x-12 y(x)+1\right )-\frac {9 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) y(x)}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}+\frac {9 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) y(x)}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}+\frac {\tanh ^{-1}\left (\frac {3 x-1}{2 \sqrt {1-9 y(x)}}\right )}{\sqrt {1-9 y(x)}}-\frac {2 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}+\frac {\tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}-\frac {2 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {1-9 y(x)} \sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}\right )=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.243 (sec), leaf count = 101

$\left \{ y \left ( x \right ) =-{\frac {{x}^{2}}{3}},y \left ( x \right ) ={\frac {-3\,{{\it \_C1}}^{2}{x}^{2}-2\,\sqrt {3}{\it \_C1}\,x+3}{12\,{{\it \_C1}}^{2}}},y \left ( x \right ) ={\frac {-3\,{{\it \_C1}}^{2}{x}^{2}+2\,\sqrt {3}{\it \_C1}\,x+3}{12\,{{\it \_C1}}^{2}}},y \left ( x \right ) =-{\frac {\sqrt {3}{\it \_C1}\,x}{6}}+{\frac {{{\it \_C1}}^{2}}{4}}-{\frac {{x}^{2}}{4}},y \left ( x \right ) ={\frac {\sqrt {3}{\it \_C1}\,x}{6}}+{\frac {{{\it \_C1}}^{2}}{4}}-{\frac {{x}^{2}}{4}} \right \}$