##### 4.6.15 $$a x^2+b x y(x)+c y(x)^2+x^2 y'(x)=0$$

ODE
$a x^2+b x y(x)+c y(x)^2+x^2 y'(x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _Riccati]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.345079 (sec), leaf count = 66

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

Maple
cpu = 0.054 (sec), leaf count = 71

$\left [y \left (x \right ) = -\frac {x \left (\sqrt {4 c a -b^{2}-2 b -1}\, \tan \left (\frac {\ln \left (x \right ) \sqrt {4 c a -b^{2}-2 b -1}}{2}+\frac {\textit {\_C1} \sqrt {4 c a -b^{2}-2 b -1}}{2}\right )+b +1\right )}{2 c}\right ]$ Mathematica raw input

DSolve[a*x^2 + b*x*y[x] + c*y[x]^2 + x^2*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/2*(x*(1 + b - Sqrt[-1 - 2*b - b^2 + 4*a*c]*Tan[(Sqrt[-1 - 2*b - b^2
 + 4*a*c]*(C[1] - Log[x]))/2]))/c}}

Maple raw input

dsolve(x^2*diff(y(x),x)+a*x^2+b*x*y(x)+c*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = -1/2*x*((4*a*c-b^2-2*b-1)^(1/2)*tan(1/2*ln(x)*(4*a*c-b^2-2*b-1)^(1/2)+1/
2*_C1*(4*a*c-b^2-2*b-1)^(1/2))+b+1)/c]