54.2.42 problem 618

Internal problem ID [11916]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 618
Date solved : Sunday, October 12, 2025 at 02:02:45 AM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.618 (sec). Leaf size: 37

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

\begin{align*} y &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1}}{x}\right )-1 \\ y &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-{\mathrm e}^{x} c_1 -1}}{x}\right )-1 \\ \end{align*}

✓ Mathematica. Time used: 60.128 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to -1-W\left (-\frac {e^{-1+c_1 e^x}}{x}\right ) \end{align*}

✓ Sympy. Time used: 1.137 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = - W\left (- \frac {e^{C_{1} e^{x} - 1}}{x}\right ) - 1 \]