23.2.109 problem 111

Internal problem ID [5464]
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 : 111
Date solved : Tuesday, September 30, 2025 at 12:43:59 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}-y y^{\prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 42

ode:=x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= -\frac {x \left (\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_1 a}\right )-1\right )^{2} a}{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_1 a}\right )} \\ \end{align*}

✓ Mathematica. Time used: 1.533 (sec). Leaf size: 173

ode=x (D[y[x],x])^2-y[x] D[y[x],x]+a y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}-4 a}}{\sqrt {\frac {y(x)}{x}}}\right )+\frac {\sqrt {\frac {y(x)}{x}-4 a}}{\sqrt {\frac {y(x)}{x}-4 a}-\sqrt {\frac {y(x)}{x}}}=\frac {\log (x)}{2}+c_1,y(x)\right ]\\ \text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}-4 a}}{\sqrt {\frac {y(x)}{x}}}\right )-\frac {\sqrt {\frac {y(x)}{x}-4 a}}{\sqrt {\frac {y(x)}{x}-4 a}+\sqrt {\frac {y(x)}{x}}}=-\frac {\log (x)}{2}+c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded