81.15.21 problem 19-22

Internal problem ID [21729]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 19. Change of variables. Page 483
Problem number : 19-22
Date solved : Thursday, October 02, 2025 at 08:01:28 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.267 (sec). Leaf size: 38

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

\begin{align*} y &= \frac {4}{4 c_1 +x} \\ y &= {\mathrm e}^{-c_1 \left (c_2 +x \right )}+c_1 \\ y &= {\mathrm e}^{c_1 \left (c_2 +x \right )}-c_1 \\ \end{align*}

✓ Mathematica. Time used: 5.091 (sec). Leaf size: 93

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

\begin{align*} y(x)&\to \frac {1}{2} \left (e^{\frac {1}{2} \left (1+e^{c_1}\right ) (x+c_2)}-1-e^{c_1}\right )\\ y(x)&\to \frac {1-e^{\frac {x+c_2}{1-\tanh \left (\frac {c_1}{2}\right )}}}{-1+\tanh \left (\frac {c_1}{2}\right )}\\ y(x)&\to \frac {1}{2} \left (-1+e^{\frac {x+c_2}{2}}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (sqrt(-108*y(x)**3*Derivative(y(x), (x, 2))**3 + 729*Derivative(y(x), (x, 2))**4)/2 - 27*Derivative(y(x), (x, 2))**2/2)**(1/3)/3 + Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2))/(sqrt(-108*y(x)**3*Derivative(y(x), (x, 2))**3 + 729*Derivative(y(x), (x, 2))**4)/2 - 27*Derivative(y(x), (x, 2))**2/2)**(1/3) cannot be solved by the factorable group method