22.1.133 problem 156

Internal problem ID [4439]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 156
Date solved : Friday, October 03, 2025 at 01:21:08 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 40

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

\[ y = \frac {-1-\operatorname {LambertW}\left (-{\mathrm e}^{-x}\right )^{2}+\left (-x +c_1 \right ) \operatorname {LambertW}\left (-{\mathrm e}^{-x}\right )}{\operatorname {LambertW}\left (-{\mathrm e}^{-x}\right )} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to -\frac {1}{W\left (-e^{-x}\right )}+\log \left (W\left (-e^{-x}\right )\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.354 (sec). Leaf size: 19

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

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