29.33.20 problem 982

Internal problem ID [5559]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 982
Date solved : Monday, January 27, 2025 at 11:48:23 AM
CAS classification : [_quadrature]

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

Solution by Maple

Time used: 0.061 (sec). Leaf size: 115

dsolve((a^2-y(x)^2)*diff(y(x),x)^2 = y(x)^2,y(x), singsol=all)
 
\begin{align*} y \left (x \right ) &= 0 \\ a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\sqrt {a^{2}-y \left (x \right )^{2}}\, \operatorname {csgn}\left (a \right )+a \right )}{y \left (x \right )}\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-\sqrt {a^{2}-y \left (x \right )^{2}}-c_{1} +x &= 0 \\ -a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\sqrt {a^{2}-y \left (x \right )^{2}}\, \operatorname {csgn}\left (a \right )+a \right )}{y \left (x \right )}\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+\sqrt {a^{2}-y \left (x \right )^{2}}-c_{1} +x &= 0 \\ \end{align*}

Solution by Mathematica

Time used: 0.336 (sec). Leaf size: 102

DSolve[(a^2-y[x]^2) (D[y[x],x])^2==y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}