61.2.21 problem 21

Internal problem ID [12027]
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 : 21
Date solved : Monday, January 27, 2025 at 11:51:31 PM
CAS classification : [_Riccati]

\begin{align*} x^{n +1} y^{\prime }&=a \,x^{2 n} y^{2}+c \,x^{m}+d \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 235

dsolve(x^(n+1)*diff(y(x),x)=a*x^(2*n)*y(x)^2+c*x^m+d,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.229 (sec). Leaf size: 1890

DSolve[x^(n+1)*D[y[x],x]==a*x^(2*n)*y[x]^2+c*x^m+d,y[x],x,IncludeSingularSolutions -> True]
 

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