54.2.113 problem 689

Internal problem ID [11987]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 689
Date solved : Tuesday, September 30, 2025 at 11:52:01 PM
CAS classification : [[_homogeneous, `class D`], _Riccati]

\begin{align*} y^{\prime }&=\frac {x y-y-{\mathrm e}^{x +1} x^{3}+{\mathrm e}^{x +1} x y^{2}}{\left (x -1\right ) x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

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

\[ y = -\tanh \left ({\mathrm e}^{x +1}-{\mathrm e}^{2} \operatorname {Ei}_{1}\left (1-x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 54

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

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

✗ Sympy

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

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