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 = 0.365254 (sec), leaf count = 268

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

Maple ✓
cpu = 0.091 (sec), leaf count = 118

\[\left [y \left (x \right ) = \frac {\sqrt {c a}\, \textit {\_C1} \BesselY \left (-\frac {1}{2}-\frac {b}{2}, \sqrt {c a}\, x \right )}{x^{2} c \left (\BesselY \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {c a}\, x \right ) \textit {\_C1} +\BesselJ \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {c a}\, x \right )\right )}+\frac {\sqrt {c a}\, \BesselJ \left (-\frac {1}{2}-\frac {b}{2}, \sqrt {c a}\, x \right )}{x^{2} c \left (\BesselY \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {c a}\, x \right ) \textit {\_C1} +\BesselJ \left (-\frac {3}{2}-\frac {b}{2}, \sqrt {c a}\, x \right )\right )}\right ]\] 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] -> -1/2*(Sqrt[a]*Sqrt[c]*x*BesselY[(1 + b)/2, Sqrt[a]*Sqrt[c]*x] + (3 + b
)*BesselY[(3 + b)/2, Sqrt[a]*Sqrt[c]*x] - Sqrt[a]*Sqrt[c]*x*BesselY[(5 + b)/2, S
qrt[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]*S
qrt[c]*x]*C[1] - Sqrt[a]*Sqrt[c]*x*BesselJ[(5 + b)/2, Sqrt[a]*Sqrt[c]*x]*C[1])/(
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))

Maple raw output

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