89.1.17 problem 17

Internal problem ID [24252]
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 : 17
Date solved : Thursday, October 02, 2025 at 10:02:04 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 2.335 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to W\left (\frac {e^{-x+c_1}}{x}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.172 (sec). Leaf size: 10

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

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