80.5.1 problem 2

Internal problem ID [18488]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 2
Date solved : Monday, March 31, 2025 at 05:37:21 PM
CAS classification : [_rational, _dAlembert]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 65

ode:=x*diff(y(x),x)^2-y(x)+2*diff(y(x),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: 12.594 (sec). Leaf size: 50

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

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

✗ 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