60.2.135 problem 711

Internal problem ID [10709]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 711
Date solved : Sunday, March 30, 2025 at 06:24:57 PM
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 )-1\right ) y}{x +1} \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 27

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

\[ y = {\mathrm e}^{{\mathrm e}^{-x} c_1 -\operatorname {Ei}_{1}\left (-x -1\right ) {\mathrm e}^{-x -1}} \]

✓ Mathematica. Time used: 0.261 (sec). Leaf size: 70

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

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

✓ Sympy. Time used: 1.637 (sec). Leaf size: 17

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

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