29.31.15 problem 914
Internal
problem
ID
[5494]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
31
Problem
number
:
914
Date
solved
:
Monday, January 27, 2025 at 11:31:38 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} \left (-x^{2}+1\right ) {y^{\prime }}^{2}&=1-y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.071 (sec). Leaf size: 162
dsolve((-x^2+1)*diff(y(x),x)^2 = 1-y(x)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -1 \\
y \left (x \right ) &= 1 \\
\frac {\sqrt {y \left (x \right )^{2}-1}\, \ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {1+y \left (x \right )}}-\frac {\int _{}^{x}\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\textit {\_a}^{2}-1}d \textit {\_a}}{\sqrt {y \left (x \right )-1}\, \sqrt {1+y \left (x \right )}}+c_{1} &= 0 \\
\frac {\sqrt {y \left (x \right )^{2}-1}\, \ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {1+y \left (x \right )}}+\frac {\int _{}^{x}\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\textit {\_a}^{2}-1}d \textit {\_a}}{\sqrt {y \left (x \right )-1}\, \sqrt {1+y \left (x \right )}}+c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.255 (sec). Leaf size: 98
DSolve[(1-x^2) (D[y[x],x])^2==1-y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{2} e^{-c_1} \left (\left (1+e^{2 c_1}\right ) x-\left (-1+e^{2 c_1}\right ) \sqrt {x^2-1}\right ) \\
y(x)\to \frac {1}{2} e^{-c_1} \left (\left (-1+e^{2 c_1}\right ) \sqrt {x^2-1}+\left (1+e^{2 c_1}\right ) x\right ) \\
y(x)\to -1 \\
y(x)\to 1 \\
\end{align*}