60.7.105 problem 1717 (book 6.126)
Internal
problem
ID
[11655]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1717
(book
6.126)
Date
solved
:
Sunday, March 30, 2025 at 08:33:34 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{\prime \prime } y+a \left ({y^{\prime }}^{2}+1\right )&=0 \end{align*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 61
ode:=diff(diff(y(x),x),x)*y(x)+a*(1+diff(y(x),x)^2) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {\textit {\_a}^{a}}{\sqrt {-\textit {\_a}^{2 a}+c_1}}d \textit {\_a} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {\textit {\_a}^{a}}{\sqrt {-\textit {\_a}^{2 a}+c_1}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.799 (sec). Leaf size: 526
ode=a*(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 {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {1-e^{2 (-c_1)} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 (-c_1)} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 (-c_1)} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {1-e^{2 (-c_1)} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 (-c_1)} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 (-c_1)} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*(Derivative(y(x), x)**2 + 1) + y(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE sqrt(-(a + y(x)*Derivative(y(x), (x, 2)))/a) + Derivative(y(x), x) cannot be solved by the factorable group method