82.2.2 problem Ex. 2

Internal problem ID [18649]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 14
Problem number : Ex. 2
Date solved : Thursday, March 13, 2025 at 12:27:20 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }+\sqrt {\frac {1-y^{2}}{-x^{2}+1}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 84

ode:=diff(y(x),x)+((1-y(x)^2)/(-x^2+1))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\sqrt {\frac {-1+y \left (x \right )^{2}}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+\frac {\sqrt {-1+y \left (x \right )^{2}}\, \ln \left (y \left (x \right )+\sqrt {-1+y \left (x \right )^{2}}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+c_{1} = 0 \]

✓ Mathematica. Time used: 0.327 (sec). Leaf size: 39

ode=D[y[x],x]+Sqrt[ (1-y[x]^2)/(1-x^2)]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\cosh \left (2 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x-1}{x+1}}}\right )-c_1\right ) \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 12.967 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt((1 - y(x)**2)/(1 - x**2)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \begin {cases} 0 & \text {for}\: y{\left (x \right )} > -1 \wedge y{\left (x \right )} < 1 \end {cases} + \log {\left (\sqrt {y^{2}{\left (x \right )} - 1} + y{\left (x \right )} \right )} + \frac {\int \sqrt {\frac {y^{2}{\left (x \right )} - 1}{x^{2} - 1}}\, dx}{\sqrt {\left (y{\left (x \right )} - 1\right ) \left (y{\left (x \right )} + 1\right )}} = C_{1} \]