54.2.112 problem 688

Internal problem ID [11986]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 688
Date solved : Tuesday, September 30, 2025 at 11:51:58 PM
CAS classification : [[_homogeneous, `class D`], _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 42

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

\[ y = \tan \left (\frac {\left (x^{2}+2 x -3\right ) {\mathrm e}^{\frac {x +1}{x -1}}}{2}+4 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 55

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

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

✗ Sympy

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

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