##### 4.6.21 $$x^2 y'(x)=a+b x y(x)+c x^2 y(x)^2$$

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

[[_homogeneous, class G], _rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.315312 (sec), leaf count = 64

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

Maple
cpu = 0.042 (sec), leaf count = 73

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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