4.8.21 \(x y'(x) \left (a+b x+c x^2\right )-y(x) \left (a+b x+c x^2\right )+x^2=y(x)^2\)

ODE
\[ x y'(x) \left (a+b x+c x^2\right )-y(x) \left (a+b x+c x^2\right )+x^2=y(x)^2 \] ODE Classification

[[_homogeneous, `class D`], _rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.119629 (sec), leaf count = 104

\[\left \{\left \{y(x)\to -\frac {x \left (\exp \left (\frac {4 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )-1\right )}{\exp \left (\frac {4 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+2 c_1\right )+1}\right \}\right \}\]

Maple
cpu = 0.01 (sec), leaf count = 50

\[ \left \{ -{\it Artanh} \left ({\frac {y \relax (x ) }{x}} \right ) -2\,{\frac {1}{\sqrt {4\,ca-{b}^{2}}}\arctan \left ({\frac {2\,cx+b}{\sqrt {4\,ca-{b}^{2}}}} \right ) }-{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-arctanh(y(x)/x)-2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))-_C1 = 0