82.8.11 problem 36-12

Internal problem ID [21882]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-12
Date solved : Thursday, October 02, 2025 at 08:03:28 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 19

ode:=x = y(x)-diff(y(x),x)^2; 
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: 9.644 (sec). Leaf size: 86

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

\begin{align*} 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 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: 2.247 (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)
 

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