54.1.552 problem 565

Internal problem ID [11866]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 565
Date solved : Friday, October 03, 2025 at 02:48:57 AM
CAS classification : [_separable]

\begin{align*} y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y&=0 \end{align*}

✓ Maple. Time used: 0.181 (sec). Leaf size: 17

ode:=y(x)*ln(diff(y(x),x))+diff(y(x),x)-ln(y(x))*y(x)-x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{\frac {\operatorname {LambertW}\left ({\mathrm e}^{x}\right ) \left (2+\operatorname {LambertW}\left ({\mathrm e}^{x}\right )\right )}{2}} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 22

ode=-(x*y[x]) - Log[y[x]]*y[x] + Log[D[y[x],x]]*y[x] + D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \exp \left (\int _1^xW\left (e^{K[1]}\right )dK[1]\right ) \end{align*}

✓ Sympy. Time used: 0.264 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} e^{\left (\frac {W\left (e^{x}\right )}{2} + 1\right ) W\left (e^{x}\right )} \]