4.271   x2y′(x) = a+ bxy(x)+ cx2y(x)2

ODE

x2y′(x) = a+ bxy(x)+ cx2y(x)2

ODE Classification

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

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0331989 (sec), leaf count = 64

((                           (                  )       ) )
||||         √ -------2--------     -∘----2c1------        || ||
{{         --− 4ac-+-b-+-2b-+-1-1-−-x-−4ac+b2+2b+1+c1-+-b+-1} }
|||| y(x) → −                     2cx                     || ||
((                                                      ) )

Maple
cpu = 0.011 (sec), leaf count = 54

{                                    (                 )   }
  ln(x)−  C1 − 2√------1--------arctan  √2cxy(x)+-b+-1-- = 0
                 4ca − b2 − 2b− 1        4ca− b2 − 2b− 1

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