29.6.25 problem 171
Internal
problem
ID
[4771]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
6
Problem
number
:
171
Date
solved
:
Monday, January 27, 2025 at 09:37:11 AM
CAS
classification
:
[_rational, [_Riccati, _special]]
\begin{align*} x y^{\prime }+a +x y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 59
dsolve(x*diff(y(x),x)+a+x*y(x)^2 = 0,y(x), singsol=all)
\[
y \left (x \right ) = \frac {\sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 289
DSolve[x D[y[x],x]+a+x y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (0,2 \sqrt {a} \sqrt {x}\right )+2 \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (2,2 \sqrt {a} \sqrt {x}\right )-i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )-i c_1 \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )+i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{4 x \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 i c_1 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )} \\
y(x)\to \frac {\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )+\operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )-\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{2 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )} \\
\end{align*}