23.4.172 problem 172
Internal
problem
ID
[6474]
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
:
172
Date
solved
:
Tuesday, September 30, 2025 at 03:01:41 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 1+{y^{\prime }}^{2}+2 y y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.079 (sec). Leaf size: 89
ode:=1+diff(y(x),x)^2+2*y(x)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
-\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: 0.467 (sec). Leaf size: 397
ode=1 + D[y[x],x]^2 + 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 [-e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )+\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-e^{2 (-c_1)} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 (-c_1)}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [e^{2 (-c_1)} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}{\sqrt {\text {$\#$1}}}\right )+\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 (-c_1)}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )+\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*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(-2*y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x