29.6.23 problem 169
Internal
problem
ID
[4769]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
6
Problem
number
:
169
Date
solved
:
Monday, January 27, 2025 at 09:37:05 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }&=a \,x^{n}+b y+c y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 164
dsolve(x*diff(y(x),x) = a*x^n+b*y(x)+c*y(x)^2,y(x), singsol=all)
\[
y \left (x \right ) = \frac {\sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-b \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}{c \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.302 (sec). Leaf size: 402
DSolve[x D[y[x],x]==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/2} \left (-2 \operatorname {BesselJ}\left (\frac {b}{n}-1,\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \left (\operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )-\operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )\right )-b c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 c \left (\operatorname {BesselJ}\left (\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )} \\
y(x)\to -\frac {-\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+b \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 c \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )} \\
\end{align*}