Internal problem ID [3426]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 6
Problem number: 170.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x y^{\prime }-y b -c y^{2}=k +a \,x^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 260
dsolve(x*diff(y(x),x) = k+a*x^n+b*y(x)+c*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\sqrt {b^{2}-4 c k}\, c_{1} -c_{1} b \right ) \operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 x^{\frac {n}{2}} \operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, c_{1} +\left (-\sqrt {b^{2}-4 c k}-b \right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 \operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, x^{\frac {n}{2}}}{2 c \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.812 (sec). Leaf size: 806
DSolve[x y'[x]==k +a x^n+b y[x]+c y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} \sqrt {c} x^n \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-\sqrt {a} \sqrt {c} x^n \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b \sqrt {x^n} \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-\sqrt {a} \sqrt {c} c_1 x^n \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (1-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} c_1 x^n \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b c_1 \sqrt {x^n} \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}{2 c \sqrt {x^n} \left (\operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )} y(x)\to -\frac {-\sqrt {a} \sqrt {c} \sqrt {x^n} \operatorname {BesselJ}\left (1-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} \sqrt {x^n} \operatorname {BesselJ}\left (-\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}{2 c \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )} \end{align*}