60.2.196 problem 772

Internal problem ID [10770]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 772
Date solved : Sunday, March 30, 2025 at 06:36:12 PM
CAS classification : [`x=_G(y,y')`]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 91

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

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

✓ Sympy. Time used: 1.030 (sec). Leaf size: 14

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

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