54.2.185 problem 762

Internal problem ID [12059]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 762
Date solved : Sunday, October 12, 2025 at 02:15:05 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 22

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

\[ y = {\mathrm e}^{\frac {x +c_1}{x}} \left (x +1\right )^{-\frac {1}{x}} \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 60

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

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

✓ Sympy. Time used: 0.670 (sec). Leaf size: 14

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

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