54.2.57 problem 633

Internal problem ID [11931]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 633
Date solved : Tuesday, September 30, 2025 at 11:44:41 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.323 (sec). Leaf size: 49

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

\[ y = \operatorname {RootOf}\left (-{\mathrm e}^{\operatorname {RootOf}\left (343 \operatorname {sech}\left (\frac {\left (4 c_1 -4 x -3 \textit {\_Z} \right ) \sqrt {7}}{6}\right )^{2}+98 \,{\mathrm e}^{\textit {\_Z}}\right )}-3+2 \textit {\_Z} +2 \textit {\_Z}^{2}\right ) {\mathrm e}^{\frac {2 x}{3}} \]

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 85

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

\[ \text {Solve}\left [7 \left (3 \log \left (-\frac {2}{3} e^{-4 x/3} y(x)^2-\frac {2}{3} e^{-2 x/3} y(x)+1\right )+4 x-9 c_1\right )=6 \sqrt {7} \text {arctanh}\left (\frac {y(x)+4 e^{2 x/3}}{\sqrt {7} \left (y(x)+e^{2 x/3}\right )}\right ),y(x)\right ] \]

✗ Sympy

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

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