23.2.354 problem 405

Internal problem ID [5709]
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 : 405
Date solved : Tuesday, September 30, 2025 at 02:01:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 1.007 (sec). Leaf size: 41

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

\[ C_{1} - a \left (\begin {cases} y{\left (x \right )} + \frac {\log {\left (x \right )}}{a} - \frac {W\left (a e^{a \left (y{\left (x \right )} + \frac {\log {\left (x \right )}}{a}\right )}\right )}{a} & \text {for}\: a \neq 0 \\y{\left (x \right )} + \frac {\log {\left (x \right )}}{a} & \text {otherwise} \end {cases}\right ) + \log {\left (x \right )} = 0 \]