60.7.91 problem 1702 (book 6.111)
Internal
problem
ID
[11641]
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
:
Sunday, March 30, 2025 at 08:32:26 PM
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: 0.478 (sec). Leaf size: 464
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 {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$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