60.2.4 problem 580

Internal problem ID [10578]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 580
Date solved : Sunday, March 30, 2025 at 06:08:14 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }&=F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x} \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 28

ode:=diff(y(x),x) = F(y(x)*exp(-b*x))*exp(b*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 203

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

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

✗ Sympy

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

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