ODE
\[ x y'(x) \left (a+b x+c x^2\right )-y(x) \left (a+b x+c x^2\right )+x^2=y(x)^2 \] ODE Classification
[[_homogeneous, `class D`], _rational, _Riccati]
Book solution method
Riccati ODE, Generalized ODE
Mathematica ✓
cpu = 0.521223 (sec), leaf count = 104
\[\left \{\left \{y(x)\to -\frac {x \left (-1+\exp \left (\frac {4 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )\right )}{1+\exp \left (\frac {4 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )}\right \}\right \}\]
Maple ✓
cpu = 0.057 (sec), leaf count = 58
\[\left [y \left (x \right ) = -\tanh \left (\frac {\textit {\_C1} \sqrt {4 c a -b^{2}}+2 \arctan \left (\frac {2 c x +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}\right ) x\right ]\] Mathematica raw input
DSolve[x^2 - (a + b*x + c*x^2)*y[x] + x*(a + b*x + c*x^2)*y'[x] == y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(((-1 + E^((4*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*C[1]))*x)/(1 + E^((4*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*C[1])))}}
Maple raw input
dsolve(x*(c*x^2+b*x+a)*diff(y(x),x)+x^2-(c*x^2+b*x+a)*y(x) = y(x)^2, y(x))
Maple raw output
[y(x) = -tanh((_C1*(4*a*c-b^2)^(1/2)+2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))/(4*a*c-b^2)^(1/2))*x]