Internal problem ID [10382]
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: 42.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-b y=c \,x^{m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 172
dsolve(x*diff(y(x),x)=a*x^(n)*y(x)^2+b*y(x)+c*x^(m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\operatorname {BesselY}\left (-\frac {b -m}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {b -m}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right ) x^{\frac {m}{2}+\frac {n}{2}} \sqrt {a c}\, x^{-n +1}}{\left (\operatorname {BesselY}\left (-\frac {b +n}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {b +n}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 1.49 (sec). Leaf size: 1321
DSolve[x*y'[x]==a*x^(n)*y[x]^2+b*y[x]+c*x^(m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) x^{m+n} \operatorname {BesselJ}\left (\frac {m-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} m x^{m+n} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-\sqrt {a} \sqrt {c} n x^{m+n} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}-(b+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {2 b+m+3 n}{2 (m+n)}}-b (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-n (m+n)^{\frac {2 (b+n)}{m+n}} \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right ) \sqrt {(m+n)^2}-\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b-m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )-\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b-m}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} m (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b+n}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )+\sqrt {a} \sqrt {c} n (m+n)^{\frac {2 (b+n)}{m+n}} x^{m+n} \operatorname {BesselJ}\left (\frac {b+n}{m+n}+1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )\right )}{2 a \sqrt {(m+n)^2} \sqrt {x^{m+n}} \left (\operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {m-b}{m+n}\right ) \left ((m+n)^2\right )^{\frac {b+n}{m+n}}+(m+n)^{\frac {2 (b+n)}{m+n}} \operatorname {BesselJ}\left (\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right ) \operatorname {Gamma}\left (\frac {b+m+2 n}{m+n}\right )\right )} y(x)\to \frac {x^{-n} \left (\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (\frac {m-b}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-(b+n) \sqrt {(m+n)^2} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )-\sqrt {a} \sqrt {c} (m+n) \sqrt {x^{m+n}} \operatorname {BesselJ}\left (-\frac {b+m+2 n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )\right )}{2 a \sqrt {(m+n)^2} \operatorname {BesselJ}\left (-\frac {b+n}{m+n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^{m+n}}}{\sqrt {(m+n)^2}}\right )} \end{align*}