69.1.122 problem 181
Internal
problem
ID
[14196]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
181
Date
solved
:
Wednesday, March 05, 2025 at 10:39:33 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }&=1+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.133 (sec). Leaf size: 55
ode:=y(x)*diff(diff(y(x),x),x) = 1+diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}+{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\
y &= \frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}+{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.971 (sec). Leaf size: 464
ode=y[x]*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 [-\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