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 = 0.0147045 (sec), leaf count = 229

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

Maple
cpu = 0.114 (sec), leaf count = 164

\[ \left \{ y \relax (x ) ={\frac {1}{c} \left (\sqrt {ca} \left ({{\sl Y}_{{\frac {b+n}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {b+n}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) {x}^{{\frac {n}{2}}}-b \left ({{\sl Y}_{{\frac {b}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {b}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) \right ) \left ({{\sl Y}_{{\frac {b}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {b}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) ^{-1}} \right \} \] 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),'implicit')

Maple raw output

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