23.2.350 problem 401

Internal problem ID [5705]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 401
Date solved : Friday, October 03, 2025 at 01:43:29 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \int _1^x\frac {W\left (e^{-a} K[1]\right )}{K[1]}dK[1]+c_1 \end{align*}

✓ Sympy. Time used: 0.235 (sec). Leaf size: 20

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

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