54.2.291 problem 870

Internal problem ID [12165]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 870
Date solved : Sunday, October 12, 2025 at 02:25:47 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 37

ode:=diff(y(x),x) = (exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x+x+x^3+x^4)*exp(y(x)/x)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (2 \ln \left (2\right )+\ln \left (3\right )-\ln \left (\frac {-3 x^{4}-4 x^{3}-12 c_1 -12 x}{x}\right )\right ) x \]

✓ Mathematica. Time used: 2.801 (sec). Leaf size: 32

ode=D[y[x],x] == (E^(y[x]/x)*(x + x/E^(y[x]/x) + x^3 + x^4 + y[x]/E^(y[x]/x)))/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \log \left (-\frac {x^3}{4}-\frac {x^2}{3}-\frac {c_1}{x}-1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4 + x**3 + x + x*exp(-y(x)/x) + y(x)*exp(-y(x)/x))*exp(y(x)/x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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