29.33.19 problem 981

Internal problem ID [5558]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 981
Date solved : Monday, January 27, 2025 at 11:48:22 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.059 (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: 0.053 (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*}