4.4.23 \(x y'(x)=a x^n+b y(x)+c y(x)^2+k\)

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

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.227298 (sec), leaf count = 589

\[\left \{\left \{y(x)\to -\frac {-\sqrt {a} \sqrt {c} c_1 x^n \Gamma \left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) J_{1-\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} c_1 x^n \Gamma \left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) J_{-\frac {n+\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} x^n \Gamma \left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) J_{\frac {\sqrt {b^2-4 c k}}{n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-\sqrt {a} \sqrt {c} x^n \Gamma \left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) J_{\frac {n+\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b c_1 \sqrt {x^n} \Gamma \left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) J_{-\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b \sqrt {x^n} \Gamma \left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) J_{\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}{2 c \sqrt {x^n} \left (c_1 \Gamma \left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) J_{-\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\Gamma \left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) J_{\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )}\right \}\right \}\]

Maple
cpu = 0.143 (sec), leaf count = 231

\[ \left \{ y \relax (x ) ={\frac {1}{2\,c} \left (2\,\sqrt {ca} \left ({{\sl Y}_{{\frac {\sqrt {{b}^{2}-4\,ck}+n}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {\sqrt {{b}^{2}-4\,ck}+n}{n}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) {x}^{n/2}- \left ({{\sl Y}_{{\frac {1}{n}\sqrt {{b}^{2}-4\,ck}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{n}\sqrt {{b}^{2}-4\,ck}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) \left (\sqrt {{b}^{2}-4\,ck}+b \right ) \right ) \left ({{\sl Y}_{{\frac {1}{n}\sqrt {{b}^{2}-4\,ck}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{n}\sqrt {{b}^{2}-4\,ck}}}\left (2\,{\frac {\sqrt {ca}{x}^{n/2}}{n}}\right )} \right ) ^{-1}} \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(-(Sqrt[a]*Sqrt[c]*x^n*BesselJ[1 - Sqrt[b^2 - 4*c*k]/n, (2*Sqrt[a]*Sq
rt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[b^2 - 4*c*k]/n]) + b*Sqrt[x^n]*BesselJ[-
(Sqrt[b^2 - 4*c*k]/n), (2*Sqrt[a]*Sqrt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[b^2 
- 4*c*k]/n] + Sqrt[a]*Sqrt[c]*x^n*BesselJ[-((Sqrt[b^2 - 4*c*k] + n)/n), (2*Sqrt[
a]*Sqrt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[b^2 - 4*c*k]/n] + Sqrt[a]*Sqrt[c]*x
^n*BesselJ[-1 + Sqrt[b^2 - 4*c*k]/n, (2*Sqrt[a]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqr
t[b^2 - 4*c*k] + n)/n] + b*Sqrt[x^n]*BesselJ[Sqrt[b^2 - 4*c*k]/n, (2*Sqrt[a]*Sqr
t[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[b^2 - 4*c*k] + n)/n] - Sqrt[a]*Sqrt[c]*x^n*Bessel
J[(Sqrt[b^2 - 4*c*k] + n)/n, (2*Sqrt[a]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[b^2 - 
4*c*k] + n)/n])/(2*c*Sqrt[x^n]*(BesselJ[-(Sqrt[b^2 - 4*c*k]/n), (2*Sqrt[a]*Sqrt[
c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[b^2 - 4*c*k]/n] + BesselJ[Sqrt[b^2 - 4*c*k]
/n, (2*Sqrt[a]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[b^2 - 4*c*k] + n)/n]))}}

Maple raw input

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

Maple raw output

y(x) = 1/2*(2*(c*a)^(1/2)*(BesselY(((b^2-4*c*k)^(1/2)+n)/n,2*(c*a)^(1/2)*x^(1/2*
n)/n)*_C1+BesselJ(((b^2-4*c*k)^(1/2)+n)/n,2*(c*a)^(1/2)*x^(1/2*n)/n))*x^(1/2*n)-
(BesselY((b^2-4*c*k)^(1/2)/n,2*(c*a)^(1/2)*x^(1/2*n)/n)*_C1+BesselJ((b^2-4*c*k)^
(1/2)/n,2*(c*a)^(1/2)*x^(1/2*n)/n))*((b^2-4*c*k)^(1/2)+b))/c/(BesselY((b^2-4*c*k
)^(1/2)/n,2*(c*a)^(1/2)*x^(1/2*n)/n)*_C1+BesselJ((b^2-4*c*k)^(1/2)/n,2*(c*a)^(1/
2)*x^(1/2*n)/n))