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
:
Tuesday, March 04, 2025 at 09:47:51 PM
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*}
✓ Maple. Time used: 0.071 (sec). Leaf size: 162
ode:=(-x^2+1)*diff(y(x),x)^2 = 1-y(x)^2;
dsolve(ode,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*}
✓ Mathematica. Time used: 0.255 (sec). Leaf size: 98
ode=(1-x^2) (D[y[x],x])^2==1-y[x]^2;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 24.545 (sec). Leaf size: 112
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - x**2)*Derivative(y(x), x)**2 + y(x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \begin {cases} 0 & \text {for}\: y{\left (x \right )} > -1 \wedge y{\left (x \right )} < 1 \end {cases} + \log {\left (\sqrt {y^{2}{\left (x \right )} - 1} + y{\left (x \right )} \right )} + \frac {\int \sqrt {\frac {y^{2}{\left (x \right )} - 1}{x^{2} - 1}}\, dx}{\sqrt {\left (y{\left (x \right )} - 1\right ) \left (y{\left (x \right )} + 1\right )}} = C_{1}, \ - \begin {cases} 0 & \text {for}\: y{\left (x \right )} > -1 \wedge y{\left (x \right )} < 1 \end {cases} + \log {\left (\sqrt {y^{2}{\left (x \right )} - 1} + y{\left (x \right )} \right )} - \frac {\int \sqrt {\frac {y^{2}{\left (x \right )} - 1}{x^{2} - 1}}\, dx}{\sqrt {\left (y{\left (x \right )} - 1\right ) \left (y{\left (x \right )} + 1\right )}} = C_{1}\right ]
\]