57.1.20 problem 20
Internal
problem
ID
[9004]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
20
Date
solved
:
Sunday, March 30, 2025 at 01:58:35 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} c y^{\prime }&=\frac {a x +b y^{2}}{r} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 91
ode:=c*diff(y(x),x) = (a*x+b*y(x)^2)/r;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} \left (\operatorname {AiryAi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right ) c_1 +\operatorname {AiryBi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )\right ) r c}{b \left (c_1 \operatorname {AiryAi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )+\operatorname {AiryBi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )\right )}
\]
✓ Mathematica. Time used: 0.225 (sec). Leaf size: 517
ode=c*D[y[x],x]==(a*x+b*y[x]^2)/r;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )} \\
y(x)\to -\frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\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,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded