3.180   ODE No. 180

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

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

Mathematica: cpu = 0.123016 (sec), leaf count = 132 \[ \left \{\left \{y(x)\to -\frac {x \left (\exp \left (\frac {4 \tan ^{-1}\left (\frac {2 a x}{\sqrt {4 a c-b^2}}+\frac {b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )-1\right )}{\exp \left (\frac {4 \tan ^{-1}\left (\frac {2 a x}{\sqrt {4 a c-b^2}}+\frac {b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )+1}\right \}\right \} \]

Maple: cpu = 0.047 (sec), leaf count = 58 \[ \left \{ y \left ( x \right ) =-\tanh \left ( {1 \left ( {\it \_C1}\, \sqrt {4\,ac-{b}^{2}}+2\,\arctan \left ( {\frac {2\,ax+b}{\sqrt {4\,ac- {b}^{2}}}} \right ) \right ) {\frac {1}{\sqrt {4\,ac-{b}^{2}}}}} \right ) x \right \} \]