83.10.19 problem 19

Internal problem ID [21988]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 19
Date solved : Thursday, October 02, 2025 at 08:21:26 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 0.642 (sec). Leaf size: 14

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

\[ C_{1} - y{\left (x \right )} + W\left (x e^{y{\left (x \right )}}\right ) = 0 \]