67.2.52 problem Problem 19(e)

Internal problem ID [13938]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 19(e)
Date solved : Monday, March 31, 2025 at 08:19:36 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (1-y\right ) y^{\prime \prime }-{y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 39

ode:=(1-y(x))*diff(diff(y(x),x),x)-diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 1 \\ y &= 1-\sqrt {2 c_1 x +2 c_2 +1} \\ y &= 1+\sqrt {2 c_1 x +2 c_2 +1} \\ \end{align*}

✓ Mathematica. Time used: 0.563 (sec). Leaf size: 49

ode=(1-y[x])*D[y[x],{x,2}]- D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to 1-\sqrt {-2 c_1 x+1-2 c_2 c_1} \\ y(x)\to 1+\sqrt {-2 c_1 x+1-2 c_2 c_1} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - y(x))*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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