23.2.98 problem 100

Internal problem ID [5453]
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 : 100
Date solved : Tuesday, September 30, 2025 at 12:43:35 PM
CAS classification : [_rational, _dAlembert]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 59

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

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

✓ Mathematica. Time used: 0.521 (sec). Leaf size: 46

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(4*x*y(x) + 1) - 1)/(2*x) cannot be solved by the factorable group method