60.1.105 problem 107

Internal problem ID [10119]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 107
Date solved : Sunday, March 30, 2025 at 03:16:59 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 240

ode:=x*diff(y(x),x)+a*x^alpha*y(x)^2+b*y(x)-c*x^beta = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c \,x^{\beta } \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )}{-x^{\frac {\alpha }{2}+\frac {\beta }{2}} \left (\operatorname {BesselY}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) \sqrt {-a c}+\left (b +\beta \right ) \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_1 +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )} \]

✓ Mathematica. Time used: 0.836 (sec). Leaf size: 1474

ode=x*D[y[x],x] + a*x^\[Alpha]*y[x]^2 + b*y[x] - c*x^\[Beta]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*x**Alpha*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*x**Alpha*y(x)**2 - b*y(x) + c*x**BETA)/x cannot be solved by the factorable group method