56.34.2 problem Ex 2

Internal problem ID [14275]
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 : Thursday, October 02, 2025 at 09:27:33 AM
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: 0.037 (sec). Leaf size: 28

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 &= -c_1 \sinh \left (\frac {c_2 +x}{c_1}\right ) \\ y &= c_1 \sinh \left (\frac {c_2 +x}{c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 60.089 (sec). Leaf size: 87

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 {i e^{-c_1} \tanh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (\sqrt {e^{2 c_1}} (x+c_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)**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