23.2.357 problem 408

Internal problem ID [5712]
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 : 408
Date solved : Tuesday, September 30, 2025 at 02:01:47 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 18

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

\begin{align*} y &= {\mathrm e}^{x} \\ y &= c_1 \left (-\ln \left (c_1 \right )+x +1\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.82 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to c_1 (x+1-\log (c_1))\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -exp(x + LambertW(-y(x)*exp(-x - 1)) + 1) + Derivative(y(x), x) cannot be solved by the factorable group method