33.1.3 problem problem 46

Internal problem ID [6021]
Book : Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section : Abel ODE with constant invariant
Problem number : problem 46
Date solved : Sunday, March 30, 2025 at 10:31:48 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 1394

ode:=diff(y(x),x)-x^a*y(x)^3+3*y(x)^2-x^(-a)*y(x)-x^(-2*a)+a*x^(-a-1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 12.748 (sec). Leaf size: 231

ode=D[y[x],x]-x^a*y[x]^3+3*y[x]^2-x^(-a)*y[x]-x^(-2*a)+a*x^(-a-1) == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to x^{-a}-\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2^{\frac {3 a+1}{a-1}} x^{a+1} \left (\frac {x^{1-a}}{1-a}\right )^{\frac {a+1}{a-1}} \Gamma \left (\frac {a+1}{1-a},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a}+\frac {e^{\frac {2 x^{1-a}}{a-1}}}{\sqrt {-\frac {2^{\frac {3 a+1}{a-1}} x^{a+1} \left (\frac {x^{1-a}}{1-a}\right )^{\frac {a+1}{a-1}} \Gamma \left (\frac {a+1}{1-a},-\frac {4 x^{1-a}}{a-1}\right )}{a-1}+c_1}} \\ y(x)\to x^{-a} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*x**(-a - 1) - x**a*y(x)**3 + 3*y(x)**2 + Derivative(y(x), x) - y(x)/x**a - 1/x**(2*a),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*x**(-a - 1) - x**a*y(x)**3 + 3*y(x)**2 + Derivative(y(x), x) - y(x)/x**a - 1/x**(2*a) cannot be solved by the lie group method