89.31.11 problem 11

Internal problem ID [24956]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 246
Problem number : 11
Date solved : Thursday, October 02, 2025 at 11:36:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 32

ode:=x*diff(y(x),x)^2-y(x)*diff(y(x),x)-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}\right )-1\right )^{2}}{\operatorname {LambertW}\left (\frac {x \,{\mathrm e}}{c_1}\right )} \\ \end{align*}

✓ Mathematica. Time used: 1.586 (sec). Leaf size: 161

ode=x*D[y[x],x]^2-y[x]*D[y[x],x]-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}}{\sqrt {\frac {y(x)}{x}}}\right )+\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}+4}-\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}}{\sqrt {\frac {y(x)}{x}}}\right )-\frac {\sqrt {\frac {y(x)}{x}+4}}{\sqrt {\frac {y(x)}{x}}+\sqrt {\frac {y(x)}{x}+4}}=-\frac {\log (x)}{2}+c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded