60.1.97 problem 99
Internal
problem
ID
[10111]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
99
Date
solved
:
Sunday, March 30, 2025 at 03:16:29 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }+a y^{2}-b y-c \,x^{\beta }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 171
ode:=x*diff(y(x),x)+a*y(x)^2-b*y(x)-c*x^beta = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\sqrt {-c a}\, \left (\operatorname {BesselY}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right )\right ) x^{\frac {\beta }{2}}+b \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-c a}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}
\]
✓ Mathematica. Time used: 0.317 (sec). Leaf size: 428
ode=x*D[y[x],x] + a*y[x]^2 - b*y[x] - c*x^\[Beta]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {-a} \sqrt {c} x^{\beta /2} \left (-2 \operatorname {BesselJ}\left (\frac {b}{\beta }-1,\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 \left (\operatorname {BesselJ}\left (1-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )-\operatorname {BesselJ}\left (-\frac {b+\beta }{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )\right )-b c_1 \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+c_1 \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )\right )} \\
y(x)\to \frac {-\sqrt {-a} \sqrt {c} x^{\beta /2} \operatorname {BesselJ}\left (1-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+\sqrt {-a} \sqrt {c} x^{\beta /2} \operatorname {BesselJ}\left (-\frac {b+\beta }{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )+b \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )}{2 a \operatorname {BesselJ}\left (-\frac {b}{\beta },\frac {2 \sqrt {-a} \sqrt {c} x^{\beta /2}}{\beta }\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
BETA = symbols("BETA")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*y(x)**2 - b*y(x) - c*x**BETA + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x)**2 + b*y(x) + c*x**BETA)/x cannot be solved by the factorable group method