61.2.15 problem 15

Internal problem ID [12021]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 15
Date solved : Monday, January 27, 2025 at 11:51:04 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 220

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

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

✓ Solution by Mathematica

Time used: 0.981 (sec). Leaf size: 1779

DSolve[x^2*D[y[x],x]==a*x^2*y[x]^2+b*x^n+c,y[x],x,IncludeSingularSolutions -> True]
 

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