23.2.82 problem 84

Internal problem ID [5437]
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 : 84
Date solved : Tuesday, September 30, 2025 at 12:43:20 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 31

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

\[ y = \frac {x^{2} \left (1+2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_1}{4}}}{4}\right )\right )}{16 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_1}{4}}}{4}\right )^{2}} \]

✓ Mathematica. Time used: 1.292 (sec). Leaf size: 138

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

\begin{align*} \text {Solve}\left [\frac {8 y(x) \text {arctanh}\left (\frac {x}{\sqrt {x^2+16 y(x)}}\right )+\frac {1}{2} x \sqrt {x^2+16 y(x)}-\frac {x^2}{2}}{8 y(x)}+\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {8 y(x) \text {arctanh}\left (\frac {x}{\sqrt {x^2+16 y(x)}}\right )+\frac {1}{2} x \sqrt {x^2+16 y(x)}+\frac {x^2}{2}}{8 y(x)}=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) + 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x/4 - sqrt(x**2 + 16*y(x))/4 + Derivative(y(x), x) cannot be solved by the factorable group method