##### 4.4.33 $$a x^2 y(x)^2+x y'(x)+2 y(x)=b$$

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

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.29528 (sec), leaf count = 101

$\left \{\left \{y(x)\to \frac {i \sqrt {b} \left (Y_1\left (-i \sqrt {a} \sqrt {b} x\right )-c_1 J_1\left (i \sqrt {a} \sqrt {b} x\right )\right )}{\sqrt {a} x \left (Y_0\left (-i \sqrt {a} \sqrt {b} x\right )+c_1 J_0\left (i \sqrt {a} \sqrt {b} x\right )\right )}\right \}\right \}$

Maple
cpu = 0.128 (sec), leaf count = 104

$\left [y \left (x \right ) = -\frac {\sqrt {-a b}\, \textit {\_C1} \BesselY \left (1, \sqrt {-a b}\, x \right )}{a x \left (\textit {\_C1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}-\frac {\BesselJ \left (1, \sqrt {-a b}\, x \right ) \sqrt {-a b}}{a x \left (\textit {\_C1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (I*Sqrt[b]*(BesselY[1, (-I)*Sqrt[a]*Sqrt[b]*x] - BesselJ[1, I*Sqrt[a]*
Sqrt[b]*x]*C[1]))/(Sqrt[a]*x*(BesselY[0, (-I)*Sqrt[a]*Sqrt[b]*x] + BesselJ[0, I*
Sqrt[a]*Sqrt[b]*x]*C[1]))}}

Maple raw input

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

Maple raw output

[y(x) = -(-a*b)^(1/2)*_C1/a/x/(_C1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1
/2)*x))*BesselY(1,(-a*b)^(1/2)*x)-BesselJ(1,(-a*b)^(1/2)*x)*(-a*b)^(1/2)/a/x/(_C
1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1/2)*x))]