54.1.399 problem 410

Internal problem ID [11713]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 410
Date solved : Tuesday, September 30, 2025 at 10:14:01 PM
CAS classification : [_rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+4 y^{\prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 67

ode:=x*diff(y(x),x)^2+4*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_1 +8 \textit {\_Z} -4 x \right )} x +4 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_1 +8 \textit {\_Z} -4 x \right )+\frac {c_1}{2}-2 x \]

✓ Mathematica. Time used: 30.305 (sec). Leaf size: 117

ode=-2*y[x] + 4*D[y[x],x] + x*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=-\frac {2 (2 K[1]-y(K[1]))}{K[1]^2},y(x)=\exp \left (\int _1^{K[1]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right ) \int _1^{K[1]}\frac {4 \exp \left (-\int _1^{K[3]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right )}{K[3]-2}dK[3]+c_1 \exp \left (\int _1^{K[1]}-\frac {4}{(K[2]-2) K[2]}dK[2]\right )\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)**2 - 2*y(x) + 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(2*x*y(x) + 4) - 2)/x cannot be solved by the factorable group method