54.1.398 problem 409

Internal problem ID [11712]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 409
Date solved : Tuesday, September 30, 2025 at 10:14:00 PM
CAS classification : [_rational, _dAlembert]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 65

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

\[ y = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_1 -2 \textit {\_Z} -x \right )} x -2 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_1 -2 \textit {\_Z} -x \right )+c_1 -x \]

✓ Mathematica. Time used: 0.907 (sec). Leaf size: 50

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

\[ \text {Solve}\left [\left \{x=\frac {2 K[1]-2 \log (K[1])}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2-2 K[1]\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(x*y(x) + 1) + 1)/x cannot be solved by the factorable group method