60.2.187 problem 763

Internal problem ID [10761]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 763
Date solved : Sunday, March 30, 2025 at 06:35:26 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 )+x \right ) y}{x \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 72

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]
 

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

✓ Sympy. Time used: 0.944 (sec). Leaf size: 17

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)
 

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