Internal problem ID [2918]
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]
Solve \begin {gather*} \boxed {y^{\prime } x -k -a \,x^{n}-b y-c y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \relax (x ) = \frac {\left (-\sqrt {b^{2}-4 c k}\, c_{1}-c_{1} b \right ) \BesselY \left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 x^{\frac {n}{2}} \sqrt {a c}\, \BesselY \left (\frac {\sqrt {b^{2}-4 c k}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\left (-\sqrt {b^{2}-4 c k}-b \right ) \BesselJ \left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 \BesselJ \left (\frac {\sqrt {b^{2}-4 c k}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, x^{\frac {n}{2}}}{2 c \left (\BesselY \left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.636 (sec). Leaf size: 602
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 {-b \sqrt {x^n} \text {Gamma}\left (\frac {\sqrt {b^2-4 c k}+n}{n}\right ) J_{\frac {\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 \left (\text {Gamma}\left (\frac {\sqrt {b^2-4 c k}+n}{n}\right ) \left (J_{\frac {n+\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-J_{\frac {\sqrt {b^2-4 c k}}{n}-1}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )+c_1 \text {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \left (J_{1-\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-J_{-\frac {n+\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )\right )-b c_1 \sqrt {x^n} \text {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 )}{2 \sqrt {x^n} \left (c \text {Gamma}\left (\frac {\sqrt {b^2-4 c k}+n}{n}\right ) J_{\frac {\sqrt {b^2-4 c k}}{n}}\left (\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+c c_1 \text {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 )\right )} \\ y(x)\to -\frac {\frac {2 n \, _0\tilde {F}_1\left (;-\frac {\sqrt {b^2-4 c k}}{n};-\frac {a c x^n}{n^2}\right )}{\, _0\tilde {F}_1\left (;1-\frac {\sqrt {b^2-4 c k}}{n};-\frac {a c x^n}{n^2}\right )}+\sqrt {b^2-4 c k}+b}{2 c} \\ \end{align*}