60.7.115 problem 1728 (book 6.137)
Internal
problem
ID
[11665]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1728
(book
6.137)
Date
solved
:
Wednesday, March 05, 2025 at 02:37:11 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 y^{\prime \prime } y+{y^{\prime }}^{2}+1&=0 \end{align*}
✓ Maple. Time used: 0.085 (sec). Leaf size: 327
ode:=2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 1.36 (sec). Leaf size: 516
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 [-\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {e^{3 (-c_1)} \sqrt {1-\text {$\#$1} e^{-2 (-c_1)}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-(-c_1)}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 (-c_1)}\right )}{\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 (-c_1)} \sqrt {1-\text {$\#$1} e^{-2 (-c_1)}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-(-c_1)}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 (-c_1)}\right )}{\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$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) cannot be solved by the factorable group method