60.1.98 problem 100
Internal
problem
ID
[10112]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
100
Date
solved
:
Sunday, March 30, 2025 at 03:16:33 PM
CAS
classification
:
[_rational, [_Riccati, _special]]
\begin{align*} x y^{\prime }+x y^{2}+a&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 59
ode:=x*diff(y(x),x)+x*y(x)^2+a = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_1 +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )}
\]
✓ Mathematica. Time used: 0.249 (sec). Leaf size: 289
ode=x*D[y[x],x] + x*y[x]^2 + a==0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a + x*y(x)**2 + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a/x + y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method