29.6.24 problem 170
Internal
problem
ID
[4770]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
6
Problem
number
:
170
Date
solved
:
Monday, January 27, 2025 at 09:37:08 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }&=k +a \,x^{n}+b y+c y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 231
dsolve(x*diff(y(x),x) = k+a*x^n+b*y(x)+c*y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {2 \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \sqrt {a c}\, x^{\frac {n}{2}}-\left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {b^{2}-4 c k}+b \right )}{2 c \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.680 (sec). Leaf size: 806
DSolve[x D[y[x],x]==k +a x^n+b y[x]+c y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {a} \sqrt {c} x^n \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n}-1,\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-\sqrt {a} \sqrt {c} x^n \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b \sqrt {x^n} \operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )-\sqrt {a} \sqrt {c} c_1 x^n \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (1-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} c_1 x^n \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b c_1 \sqrt {x^n} \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}{2 c \sqrt {x^n} \left (\operatorname {Gamma}\left (\frac {n+\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {b^2-4 c k}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )\right )} \\
y(x)\to -\frac {-\sqrt {a} \sqrt {c} \sqrt {x^n} \operatorname {BesselJ}\left (1-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+b \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )+\sqrt {a} \sqrt {c} \sqrt {x^n} \operatorname {BesselJ}\left (-\frac {n+\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )}{2 c \operatorname {BesselJ}\left (-\frac {\sqrt {b^2-4 c k}}{n},\frac {2 \sqrt {a} \sqrt {c} \sqrt {x^n}}{n}\right )} \\
\end{align*}