54.2.117 problem 693

Internal problem ID [11991]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 693
Date solved : Tuesday, September 30, 2025 at 11:52:16 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

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

✓ Maple. Time used: 0.076 (sec). Leaf size: 33

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

\[ y = \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}-\textit {\_a} b +1}d \textit {\_a} +c_1 \right ) {\mathrm e}^{b x} \]

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 100

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

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}}\frac {1}{K[1]^3-\frac {3 (3 b+1) K[1]}{(9 b+29)^{2/3}}+1}dK[1]=\frac {1}{9} x e^{2 b x} \left ((9 b+29) e^{-3 b x}\right )^{2/3}+c_1,y(x)\right ] \]

✗ Sympy

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

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