83.6.5 problem 2 (e)

Internal problem ID [21948]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VII at page 50
Problem number : 2 (e)
Date solved : Thursday, October 02, 2025 at 08:19:03 PM
CAS classification : [[_1st_order, _with_exponential_symmetries]]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 27

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-x -\textit {\_Z} -\operatorname {Ei}_{1}\left ({\mathrm e}^{\textit {\_Z}}\right ) {\mathrm e}^{{\mathrm e}^{\textit {\_Z}}}+{\mathrm e}^{{\mathrm e}^{\textit {\_Z}}} c_1 \right )} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 35

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

\[ \text {Solve}\left [x=e^{y(x)} \left (\operatorname {ExpIntegralEi}(-y(x))-e^{-y(x)} \log (y(x))\right )+c_1 e^{y(x)},y(x)\right ] \]

✓ Sympy. Time used: 1.302 (sec). Leaf size: 792

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + 2*y(x) - log(y(x))*Derivative(y(x), x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]