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 = 0.00910832 (sec), leaf count = 143

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

Maple
cpu = 0.111 (sec), leaf count = 119

\[ \left \{ y \relax (x ) =-2\,{\frac {a \left ({{\sl J}_{1}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )}{\it \_C1}+{{\sl Y}_{1}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )} \right ) }{\sqrt {x} \left (-2\,{{\sl J}_{0}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )}\sqrt {a}\sqrt [4]{x}\sqrt {b}{\it \_C1}-2\,{{\sl Y}_{0}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )}\sqrt {a}\sqrt {b}\sqrt [4]{x}+{{\sl J}_{1}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )}{\it \_C1}+{{\sl Y}_{1}\left (4\,\sqrt {a}\sqrt {b}\sqrt [4]{x}\right )} \right ) }} \right \} \] 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),'implicit')

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)*x^(1/4)*b^(
1/2)*_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)))