4.6.19 \(x^2 y'(x)=a+b x^2 y(x)^2\)

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

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

Book solution method
Riccati ODE, Main form

Mathematica
cpu = 0.00671612 (sec), leaf count = 51

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

Maple
cpu = 0.012 (sec), leaf count = 37

\[ \left \{ \ln \relax (x ) -{\it \_C1}-2\,{\frac {1}{\sqrt {4\,ab-1}}\arctan \left ({\frac {2\,bxy \relax (x ) +1}{\sqrt {4\,ab-1}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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