54.7.91 problem 1702 (book 6.111)

Internal problem ID [12940]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1702 (book 6.111)
Date solved : Wednesday, October 01, 2025 at 02:46:32 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.284 (sec). Leaf size: 396

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {\text {arctanh}\left (\frac {\text {$\#$1} \sqrt {e^{2 c_1}}}{\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\right )}{\sqrt {e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {arctanh}\left (\frac {\text {$\#$1} \sqrt {e^{2 (-c_1)}}}{\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\right )}{\sqrt {e^{2 (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {\text {arctanh}\left (\frac {\text {$\#$1} \sqrt {e^{2 c_1}}}{\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\right )}{\sqrt {e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][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