83.23.19 problem 19

Internal problem ID [19300]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 01:25:14 PM
CAS classification : [_quadrature]

\begin{align*} \left (1-y^{2}\right ) {y^{\prime }}^{2}&=1 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 46

dsolve((1-y(x)^2)*diff(y(x),x)^2=1,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*}

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 69

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