[_rational, _Riccati]
Book solution method
Riccati ODE, Generalized ODE
Mathematica ✓
cpu = 3.17175 (sec), leaf count = 268
Maple ✓
cpu = 0.088 (sec), leaf count = 76
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))