56.34.4 problem Ex 4

Internal problem ID [14277]
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 4
Date solved : Thursday, October 02, 2025 at 09:27:38 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 20

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

\begin{align*} y &= 0 \\ y &= \frac {{\mathrm e}^{c_1 \left (c_2 +x \right )}-2}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.903 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \frac {-2+e^{c_1 (x+c_2)}}{c_1}\\ y(x)&\to \text {Indeterminate} \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)**2 + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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