54.2.405 problem 984

Internal problem ID [12279]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 984
Date solved : Sunday, October 12, 2025 at 02:28:11 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {y \left (x^{2} y^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (x -1\right )}{x} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 44

ode:=diff(y(x),x) = y(x)/x*(x^2*y(x)^2+y(x)*x*exp(x)+exp(x)^2)/exp(x)^2*(x-1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (x \left ({\mathrm e}^{\textit {\_Z}}+9\right )\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} x +9\right )+x}}{9 x} \]

✓ Mathematica. Time used: 0.43 (sec). Leaf size: 112

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

\[ \text {Solve}\left [\int _1^{\frac {e^{-x} (x-1)+3 e^{-2 x} x y(x) (x-1)}{\sqrt [3]{2} \sqrt [3]{e^{-3 x} (x-1)^3}}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]=\frac {2^{2/3} e^{-x} (x-1) (x-\log (x))}{9 \sqrt [3]{e^{-3 x} (x-1)^3}}+c_1,y(x)\right ] \]

✗ Sympy

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

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