54.2.286 problem 865

Internal problem ID [12160]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 865
Date solved : Sunday, October 12, 2025 at 02:25:02 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\left (\frac {\ln \left (-1+y\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (-1+y\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right ) \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 19

ode:=diff(y(x),x) = (1/(1-y(x))/ln(x)/x*ln(y(x)-1)*y(x)-1/(1-y(x))/ln(x)/x*ln(y(x)-1)-f(x))*(1-y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{c_1 +\int \frac {f \left (x \right )}{\ln \left (x \right )}d x}+1 \]

✓ Mathematica. Time used: 0.223 (sec). Leaf size: 87

ode=D[y[x],x] == (1 - y[x])*(-f[x] - Log[-1 + y[x]]/(x*Log[x]*(1 - y[x])) + (Log[-1 + y[x]]*y[x])/(x*Log[x]*(1 - y[x]))); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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