89.1.19 problem 19

Internal problem ID [24254]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:02:08 PM
CAS classification : [_separable]

\begin{align*} 1+\ln \left (x \right )+\left (1+\ln \left (y\right )\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 24

ode:=1+ln(x)+(1+ln(y(x)))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x \ln \left (x \right )+c_1}{\operatorname {LambertW}\left (-x \ln \left (x \right )-c_1 \right )} \]

✓ Mathematica. Time used: 60.053 (sec). Leaf size: 25

ode=(1+Log[x])+(1+Log[y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-x \log (x)+c_1}{W(-x \log (x)+c_1)} \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 12

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

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