##### 4.4.22 $$x y'(x)=a x^n+b y(x)+c y(x)^2$$

ODE
$x y'(x)=a x^n+b y(x)+c y(x)^2$ ODE Classiﬁcation

[_rational, _Riccati]

Book solution method
Riccati ODE, Special cases

Mathematica
cpu = 0.344758 (sec), leaf count = 229

$\left \{\left \{y(x)\to \frac {\sqrt {a} \sqrt {c} x^{n/2} \left (-2 J_{\frac {b}{n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \left (J_{1-\frac {b}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )-J_{-\frac {b+n}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )\right )-b c_1 J_{-\frac {b}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 c \left (J_{\frac {b}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 J_{-\frac {b}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )}\right \}\right \}$

Maple
cpu = 0.099 (sec), leaf count = 225

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

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

Mathematica raw output

{{y[x] -> (-(b*BesselJ[-(b/n), (2*Sqrt[a]*Sqrt[c]*x^(n/2))/n]*C[1]) + Sqrt[a]*Sq
rt[c]*x^(n/2)*(-2*BesselJ[-1 + b/n, (2*Sqrt[a]*Sqrt[c]*x^(n/2))/n] + (BesselJ[1
- b/n, (2*Sqrt[a]*Sqrt[c]*x^(n/2))/n] - BesselJ[-((b + n)/n), (2*Sqrt[a]*Sqrt[c]
*x^(n/2))/n])*C[1]))/(2*c*(BesselJ[b/n, (2*Sqrt[a]*Sqrt[c]*x^(n/2))/n] + BesselJ
[-(b/n), (2*Sqrt[a]*Sqrt[c]*x^(n/2))/n]*C[1]))}}

Maple raw input

dsolve(x*diff(y(x),x) = a*x^n+b*y(x)+c*y(x)^2, y(x))

Maple raw output

[y(x) = x^(1/2*n)*(c*a)^(1/2)*_C1/c/(BesselY(b/n,2*(c*a)^(1/2)*x^(1/2*n)/n)*_C1+
BesselJ(b/n,2*(c*a)^(1/2)*x^(1/2*n)/n))*BesselY((b+n)/n,2*(c*a)^(1/2)*x^(1/2*n)/
n)+(BesselJ((b+n)/n,2*(c*a)^(1/2)*x^(1/2*n)/n)*(c*a)^(1/2)*x^(1/2*n)-BesselY(b/n
,2*(c*a)^(1/2)*x^(1/2*n)/n)*_C1*b-b*BesselJ(b/n,2*(c*a)^(1/2)*x^(1/2*n)/n))/c/(B
esselY(b/n,2*(c*a)^(1/2)*x^(1/2*n)/n)*_C1+BesselJ(b/n,2*(c*a)^(1/2)*x^(1/2*n)/n)
)]