Internal problem ID [8072]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 492.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{2}-a^{2}\right ) \left (y^{\prime }\right )^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 126
dsolve((y(x)^2-a^2)*diff(y(x),x)^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ x -\sqrt {a^{2}-y \relax (x )^{2}}+\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-y \relax (x )^{2}}}{y \relax (x )}\right )}{\sqrt {a^{2}}}-c_{1} = 0 \\ x +\sqrt {a^{2}-y \relax (x )^{2}}-\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-y \relax (x )^{2}}}{y \relax (x )}\right )}{\sqrt {a^{2}}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.327 (sec). Leaf size: 102
DSolve[y[x]^2 + (-a^2 + y[x]^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \tanh ^{-1}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \tanh ^{-1}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}