66.1.10 problem Problem 10

Internal problem ID [13786]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 10
Date solved : Monday, March 31, 2025 at 08:11:51 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.192 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 1.112 (sec). Leaf size: 14

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

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