54.1.14 problem 14
Internal
problem
ID
[11328]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
14
Date
solved
:
Tuesday, September 30, 2025 at 07:42:23 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }+y^{2}+a \,x^{m}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 187
ode:=diff(y(x),x)+y(x)^2+a*x^m = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\operatorname {BesselJ}\left (\frac {3+m}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) \sqrt {a}\, x^{\frac {m}{2}+1} c_1 -\operatorname {BesselY}\left (\frac {3+m}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right ) \sqrt {a}\, x^{\frac {m}{2}+1}+c_1 \operatorname {BesselJ}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )+\operatorname {BesselY}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )}{x \left (c_1 \operatorname {BesselJ}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )+\operatorname {BesselY}\left (\frac {1}{m +2}, \frac {2 \sqrt {a}\, x^{\frac {m}{2}+1}}{m +2}\right )\right )}
\]
✓ Mathematica. Time used: 0.22 (sec). Leaf size: 517
ode=D[y[x],x] + y[x]^2 + a*x^m==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt {a} x^{\frac {m}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {1}{m+2}-1,\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )-\sqrt {a} x^{\frac {m}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (1+\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+\operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )-\sqrt {a} c_1 x^{\frac {m}{2}+1} \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {m+1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+\sqrt {a} c_1 x^{\frac {m}{2}+1} \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (-\frac {m+3}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+c_1 \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )}{2 x \left (\operatorname {Gamma}\left (1+\frac {1}{m+2}\right ) \operatorname {BesselJ}\left (\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )+c_1 \operatorname {Gamma}\left (\frac {m+1}{m+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )\right )}\\ y(x)&\to \frac {1}{2} \left (\frac {\sqrt {a} x^{m/2} \left (\operatorname {BesselJ}\left (-\frac {m+3}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )-\operatorname {BesselJ}\left (\frac {m+1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{m+2},\frac {2 \sqrt {a} x^{\frac {m}{2}+1}}{m+2}\right )}+\frac {1}{x}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
m = symbols("m")
y = Function("y")
ode = Eq(a*x**m + y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a*x**m + y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method