23.4.189 problem 189
Internal
problem
ID
[6491]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
189
Date
solved
:
Tuesday, September 30, 2025 at 03:01:57 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 y y^{\prime \prime }&={y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right ) \end{align*}
✓ Maple. Time used: 0.079 (sec). Leaf size: 93
ode:=2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2*(1+diff(y(x),x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
-\sqrt {y \left (c_1 -y\right )}+\frac {c_1 \arctan \left (\frac {-2 y+c_1}{2 \sqrt {y \left (c_1 -y\right )}}\right )}{2}-x -c_2 &= 0 \\
\sqrt {y \left (c_1 -y\right )}-\frac {c_1 \arctan \left (\frac {-2 y+c_1}{2 \sqrt {y \left (c_1 -y\right )}}\right )}{2}-x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.882 (sec). Leaf size: 489
ode=2*y[x]*D[y[x],{x,2}] == D[y[x],x]^2*(1 + D[y[x],x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{c_1}}{\sqrt {-1+\text {$\#$1} e^{2 c_1}}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{c_1}}{\sqrt {-1+\text {$\#$1} e^{2 c_1}}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-i e^{-(-c_1)} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}-e^{-(-c_1)} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{-c_1}}{\sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-(-c_1)} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}-e^{-(-c_1)} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{-c_1}}{\sqrt {-1+\text {$\#$1} e^{2 (-c_1)}}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{c_1}}{\sqrt {-1+\text {$\#$1} e^{2 c_1}}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {\text {$\#$1}} \sqrt {-1+\text {$\#$1} e^{2 c_1}}-e^{-c_1} \text {arctanh}\left (\frac {\sqrt {\text {$\#$1}} e^{c_1}}{\sqrt {-1+\text {$\#$1} e^{2 c_1}}}\right )\right )\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(Derivative(y(x), x)**2 + 1)*Derivative(y(x), x)**2 + 2*y(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(sqrt(8*y(x)*Derivative(y(x), (x, 2)) + 1)/2 - 1/2) + Deriv