23.2.3 problem 3

Internal problem ID [5358]
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 : 3
Date solved : Tuesday, September 30, 2025 at 12:36:06 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 9.52 (sec). Leaf size: 98

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

\begin{align*} y(x)&\to -W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2-2 W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )+x-1\\ y(x)&\to -W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2-2 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right )+x-1\\ y(x)&\to x-1 \end{align*}

✓ Sympy. Time used: 1.144 (sec). Leaf size: 54

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

\[ \left [ C_{1} + x - 2 \sqrt {x - y{\left (x \right )}} + 2 \log {\left (\sqrt {x - y{\left (x \right )}} + 1 \right )} = 0, \ C_{1} + x + 2 \sqrt {x - y{\left (x \right )}} + 2 \log {\left (\sqrt {x - y{\left (x \right )}} - 1 \right )} = 0\right ] \]