56.34.1 problem Ex 1

Internal problem ID [14274]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 58. Independent variable absent. Page 135
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:27:33 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 21

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

\begin{align*} y &= 0 \\ y &= \frac {c_1}{{\mathrm e}^{-\left (c_2 +x \right ) c_1}-1} \\ \end{align*}

✓ Mathematica. Time used: 0.868 (sec). Leaf size: 43

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2*Derivative(y(x), x) + 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)**3 + 4*Derivative(y(x), (x, 2)))*y(x))/2 + y(x)**2/2 + Derivative(y(x), x) cannot be solved by the factorable group method