29.9.23 problem 263

Internal problem ID [4863]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 9
Problem number : 263
Date solved : Monday, January 27, 2025 at 09:45:28 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{2} y^{\prime }&=a +b \,x^{n}+x^{2} y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 196

dsolve(x^2*diff(y(x),x) = a+b*x^n+x^2*y(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {2 \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}+1, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\sqrt {1-4 a}+1\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )}{2 x \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 a}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.870 (sec). Leaf size: 1434

DSolve[x^2 D[y[x],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*}