Internal problem ID [3519]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 9
Problem number: 263.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=a +b \,x^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 275
dsolve(x^2*diff(y(x),x) = a+b*x^n+x^2*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{\frac {n}{2}} \sqrt {b}\, c_{1} \operatorname {BesselY}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )}{x \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )}+\frac {\left (-\sqrt {1-4 a}\, c_{1} -c_{1} \right ) \operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )+2 \operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}+n}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {b}\, x^{\frac {n}{2}}+\left (-\sqrt {1-4 a}-1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )}{2 x \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.976 (sec). Leaf size: 1434
DSolve[x^2 y'[x]==a+b x^n + x^2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}}+n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2}+1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}}-i \sqrt {4 a-1} n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}-n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}+n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} \sqrt {(1-4 a) n^2} \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) b^{\frac {i \sqrt {4 a-1}}{n}}-n^{\frac {2 i \sqrt {4 a-1}}{n}} \left (-i \sqrt {4 a-1} n+n+\sqrt {(1-4 a) n^2}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}}-n^{\frac {2 i \sqrt {4 a-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+1} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}}+n^{\frac {2 i \sqrt {4 a-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+1} \operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}+\frac {1}{2}}}{2 n x \sqrt {x^n} \left (b^{\frac {i \sqrt {4 a-1}}{n}} n^{\frac {2 \sqrt {(1-4 a) n^2}}{n^2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a}}{n}\right ) \left (x^n\right )^{\frac {i \sqrt {4 a-1}}{n}}+b^{\frac {\sqrt {(1-4 a) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a-1}}{n}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a) n^2}}{n^2}}\right )} y(x)\to \frac {\frac {\sqrt {b} \sqrt {x^n} \left (\operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )-\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )\right )}{\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )}-\frac {\sqrt {(1-4 a) n^2}}{n}+i \sqrt {4 a-1}-1}{2 x} \end{align*}