62.34.2 problem Ex 2

Internal problem ID [12915]
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 2
Date solved : Wednesday, March 05, 2025 at 08:51:25 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}+1&=0 \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 59

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

\begin{align*} y &= -\frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ y &= \frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 14.308 (sec). Leaf size: 84

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

\begin{align*} y(x)\to -\frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to \frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to -x-c_2 \\ y(x)\to x+c_2 \\ \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 + 1,0) 
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) cannot be solved by the factorable group method