51.1.22 problem 22
Internal
problem
ID
[10292]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
22
Date
solved
:
Tuesday, September 30, 2025 at 07:18:15 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} c y^{\prime }&=\frac {a x +b y^{2}}{r \,x^{2}} \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 106
ode:=c*diff(y(x),x) = (a*x+b*y(x)^2)/r/x^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {a \left (\operatorname {BesselY}\left (0, 2 \sqrt {\frac {a b}{x \,r^{2} c^{2}}}\right ) c_1 +\operatorname {BesselJ}\left (0, 2 \sqrt {\frac {a b}{x \,r^{2} c^{2}}}\right )\right )}{c r \sqrt {\frac {a b}{x \,r^{2} c^{2}}}\, \left (c_1 \operatorname {BesselY}\left (1, 2 \sqrt {\frac {a b}{x \,r^{2} c^{2}}}\right )+\operatorname {BesselJ}\left (1, 2 \sqrt {\frac {a b}{x \,r^{2} c^{2}}}\right )\right )}
\]
✓ Mathematica. Time used: 0.235 (sec). Leaf size: 492
ode=c*D[y[x],x]==(a*x+b*y[x]^2)/(r*x^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2 \sqrt {a} \sqrt {b} \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )+\frac {2 c r \operatorname {BesselY}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )}{\sqrt {\frac {1}{x}}}-2 \sqrt {a} \sqrt {b} \operatorname {BesselY}\left (2,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )-i \sqrt {a} \sqrt {b} c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )-\frac {i c c_1 r \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )}{\sqrt {\frac {1}{x}}}+i \sqrt {a} \sqrt {b} c_1 \operatorname {BesselJ}\left (2,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )}{2 b \sqrt {\frac {1}{x}} \left (2 \operatorname {BesselY}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )-i c_1 \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )\right )}\\ y(x)&\to \frac {x \left (\sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}} \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )+c r \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )-\sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}} \operatorname {BesselJ}\left (2,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )\right )}{2 b \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b} \sqrt {\frac {1}{x}}}{c r}\right )} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
r = symbols("r")
y = Function("y")
ode = Eq(c*Derivative(y(x), x) - (a*x + b*y(x)**2)/(r*x**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded