61.2.42 problem 42
Internal
problem
ID
[12048]
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
Date
solved
:
Monday, January 27, 2025 at 11:53:32 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }&=a \,x^{n} y^{2}+b y+c \,x^{m} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 166
dsolve(x*diff(y(x),x)=a*x^(n)*y(x)^2+b*y(x)+c*x^(m),y(x), singsol=all)
\[
y = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.769 (sec). Leaf size: 1578
DSolve[x*D[y[x],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 )} \\
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*}