problem 267

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

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

\begin{align*} y(x)\to \frac {-b^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}}+b^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+1} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1,\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}}-b^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}-i b^{\frac {i \sqrt {4 a c-1}}{n}} \sqrt {4 a c-1} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}+b^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}} \sqrt {(1-4 a c) n^2} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}-b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}} \left (-i \sqrt {4 a c-1} n+n+\sqrt {(1-4 a c) n^2}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}-b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}}+b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1} \operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}}}{2 c n x \sqrt {x^n} \left (b^{\frac {i \sqrt {4 a c-1}}{n}} c^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}} \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 a c}}{n}\right ) \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}}+b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}} \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 a c}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\right )} \\ y(x)\to \frac {\frac {\sqrt {b} \sqrt {c} \sqrt {x^n} \left (\operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )-\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )\right )}{\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )}-\frac {\sqrt {n^2 (1-4 a c)}}{n}+i \sqrt {4 a c-1}-1}{2 c x} \\ y(x)\to \frac {\frac {\sqrt {b} \sqrt {c} \sqrt {x^n} \left (\operatorname {BesselJ}\left (1-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )-\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1,\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )\right )}{\operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 a c) n^2}}{n^2},\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right )}-\frac {\sqrt {n^2 (1-4 a c)}}{n}+i \sqrt {4 a c-1}-1}{2 c x} \\ \end{align*}

29.10.1 problem 267