2.525   ODE No. 525

$-a x y(x) y'(x)+2 a y(x)^2+y'(x)^2=0$ Mathematica : cpu = 0.0737801 (sec), leaf count = 135

$\left \{\left \{y(x)\to c_1 \exp \left (\frac {1}{2} \left (\frac {a x^2}{2}+\frac {1}{2} \sqrt {a} x \sqrt {a x^2-8}-4 \log \left (\sqrt {a} \sqrt {a x^2-8}+a x\right )\right )\right )\right \},\left \{y(x)\to c_1 \exp \left (\frac {1}{2} \left (\frac {a x^2}{2}-\frac {1}{2} \sqrt {a} x \sqrt {a x^2-8}+4 \log \left (\sqrt {a} \sqrt {a x^2-8}+a x\right )\right )\right )\right \}\right \}$ Maple : cpu = 0.035 (sec), leaf count = 122

$\left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {{a}^{2}x{\frac {1}{\sqrt {{a}^{2}}}}}+\sqrt {{a}^{2}{x}^{2}-8\,a} \right ) ^{-2\,{\frac {a}{\sqrt {{a}^{2}}}}}{{\rm e}^{{\frac {x}{4} \left ( ax+\sqrt {{a}^{2}{x}^{2}-8\,a} \right ) }}},y \left ( x \right ) ={\it \_C1}\, \left ( {{a}^{2}x{\frac {1}{\sqrt {{a}^{2}}}}}+\sqrt {{a}^{2}{x}^{2}-8\,a} \right ) ^{2\,{\frac {a}{\sqrt {{a}^{2}}}}}{{\rm e}^{-{\frac {x}{4}\sqrt {{a}^{2}{x}^{2}-8\,a}}+{\frac {a{x}^{2}}{4}}}} \right \}$