4.6.22 \(x^2 y'(x)=a+b x y(x)+c x^4 y(x)^2\)

ODE
\[ x^2 y'(x)=a+b x y(x)+c x^4 y(x)^2 \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 3.17175 (sec), leaf count = 268

\[\left \{\left \{y(x)\to -\frac {\sqrt {a} \sqrt {c} c_1 x J_{\frac {b+1}{2}}\left (\sqrt {a} \sqrt {c} x\right )+b c_1 J_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )+3 c_1 J_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} c_1 x J_{\frac {b+5}{2}}\left (\sqrt {a} \sqrt {c} x\right )+\sqrt {a} \sqrt {c} x Y_{\frac {b+1}{2}}\left (\sqrt {a} \sqrt {c} x\right )+(b+3) Y_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x Y_{\frac {b+5}{2}}\left (\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \left (c_1 J_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )+Y_{\frac {b+3}{2}}\left (\sqrt {a} \sqrt {c} x\right )\right )}\right \}\right \}\]

Maple
cpu = 0.088 (sec), leaf count = 76

\[ \left \{ y \relax (x ) ={\frac {1}{c{x}^{2}}\sqrt {ca} \left ({{\sl Y}_{-{\frac {1}{2}}-{\frac {b}{2}}}\left (\sqrt {ca}x\right )}{\it \_C1}+{{\sl J}_{-{\frac {1}{2}}-{\frac {b}{2}}}\left (\sqrt {ca}x\right )} \right ) \left ({{\sl Y}_{-{\frac {3}{2}}-{\frac {b}{2}}}\left (\sqrt {ca}x\right )}{\it \_C1}+{{\sl J}_{-{\frac {3}{2}}-{\frac {b}{2}}}\left (\sqrt {ca}x\right )} \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[x^2*y'[x] == a + b*x*y[x] + c*x^4*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[a]*Sqrt[c]*x*BesselY[(1 + b)/2, Sqrt[a]*Sqrt[c]*x] + (3 + b)*Be
sselY[(3 + b)/2, Sqrt[a]*Sqrt[c]*x] - Sqrt[a]*Sqrt[c]*x*BesselY[(5 + b)/2, Sqrt[
a]*Sqrt[c]*x] + Sqrt[a]*Sqrt[c]*x*BesselJ[(1 + b)/2, Sqrt[a]*Sqrt[c]*x]*C[1] + 3
*BesselJ[(3 + b)/2, Sqrt[a]*Sqrt[c]*x]*C[1] + b*BesselJ[(3 + b)/2, Sqrt[a]*Sqrt[
c]*x]*C[1] - Sqrt[a]*Sqrt[c]*x*BesselJ[(5 + b)/2, Sqrt[a]*Sqrt[c]*x]*C[1])/(2*c*
x^3*(BesselY[(3 + b)/2, Sqrt[a]*Sqrt[c]*x] + BesselJ[(3 + b)/2, Sqrt[a]*Sqrt[c]*
x]*C[1]))}}

Maple raw input

dsolve(x^2*diff(y(x),x) = a+b*x*y(x)+c*x^4*y(x)^2, y(x),'implicit')

Maple raw output

y(x) = (c*a)^(1/2)/x^2*(BesselY(-1/2-1/2*b,(c*a)^(1/2)*x)*_C1+BesselJ(-1/2-1/2*b
,(c*a)^(1/2)*x))/c/(BesselY(-3/2-1/2*b,(c*a)^(1/2)*x)*_C1+BesselJ(-3/2-1/2*b,(c*
a)^(1/2)*x))