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 = 0.0331989 (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 = 0.011 (sec), leaf count = 54

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

Maple raw output

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