29.33.19 problem 981
Internal
problem
ID
[5558]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
981
Date
solved
:
Tuesday, March 04, 2025 at 10:03:18 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (1-y^{2}\right ) {y^{\prime }}^{2}&=1 \end{align*}
✓ Maple. Time used: 0.059 (sec). Leaf size: 46
ode:=(1-y(x)^2)*diff(y(x),x)^2 = 1;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\
y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )-\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.053 (sec). Leaf size: 69
ode=(1-y[x]^2) (D[y[x],x])^2==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\arcsin (\text {$\#$1})\right )\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\arcsin (\text {$\#$1})\right )\&\right ][x+c_1] \\
\end{align*}
✓ Sympy. Time used: 1.413 (sec). Leaf size: 37
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - y(x)**2)*Derivative(y(x), x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {1}{y^{2} - 1}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \frac {1}{y^{2} - 1}}}\, dy = C_{1} + x\right ]
\]