##### 4.9.2 $$x^{3/2} y'(x)=a+b x^{3/2} y(x)^2$$

ODE
$x^{3/2} y'(x)=a+b x^{3/2} y(x)^2$ ODE Classiﬁcation

[_rational, [_Riccati, _special]]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 0.265972 (sec), leaf count = 143

$\left \{\left \{y(x)\to \frac {c_1 \left (\sqrt {a} \sqrt {b} \sqrt [4]{x} J_3\left (4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )-J_2\left (4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )\right )-\sqrt {a} \sqrt {b} (2+c_1) \sqrt [4]{x} J_1\left (4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}{2 b (1+c_1) x J_2\left (4 \sqrt {a} \sqrt {b} \sqrt [4]{x}\right )}\right \}\right \}$

Maple
cpu = 0.159 (sec), leaf count = 119

$\left [y \left (x \right ) = -\frac {2 a \left (\BesselJ \left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) \textit {\_C1} +\BesselY \left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right )\right )}{\sqrt {x}\, \left (-2 \BesselJ \left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}} \textit {\_C1} -2 \BesselY \left (0, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}+\BesselJ \left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right ) \textit {\_C1} +\BesselY \left (1, 4 \sqrt {a}\, \sqrt {b}\, x^{\frac {1}{4}}\right )\right )}\right ]$ Mathematica raw input

DSolve[x^(3/2)*y'[x] == a + b*x^(3/2)*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> ((-BesselJ[2, 4*Sqrt[a]*Sqrt[b]*x^(1/4)] + Sqrt[a]*Sqrt[b]*x^(1/4)*Bes
selJ[3, 4*Sqrt[a]*Sqrt[b]*x^(1/4)])*C[1] - Sqrt[a]*Sqrt[b]*x^(1/4)*BesselJ[1, 4*
Sqrt[a]*Sqrt[b]*x^(1/4)]*(2 + C[1]))/(2*b*x*BesselJ[2, 4*Sqrt[a]*Sqrt[b]*x^(1/4)
]*(1 + C[1]))}}

Maple raw input

dsolve(x^(3/2)*diff(y(x),x) = a+b*x^(3/2)*y(x)^2, y(x))

Maple raw output

[y(x) = -2*a/x^(1/2)*(BesselJ(1,4*a^(1/2)*b^(1/2)*x^(1/4))*_C1+BesselY(1,4*a^(1/
2)*b^(1/2)*x^(1/4)))/(-2*BesselJ(0,4*a^(1/2)*b^(1/2)*x^(1/4))*a^(1/2)*b^(1/2)*x^
(1/4)*_C1-2*BesselY(0,4*a^(1/2)*b^(1/2)*x^(1/4))*a^(1/2)*b^(1/2)*x^(1/4)+BesselJ
(1,4*a^(1/2)*b^(1/2)*x^(1/4))*_C1+BesselY(1,4*a^(1/2)*b^(1/2)*x^(1/4)))]