82.2.2 problem Ex. 2

Internal problem ID [18728]
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 : Tuesday, January 28, 2025 at 12:12:32 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]+Sqrt[ (1-y[x]^2)/(1-x^2)]==0,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*}