ODE
\[ \left (a^2-y(x)^2\right ) y'(x)^2=y(x)^2 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.272359 (sec), leaf count = 97
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \tanh ^{-1}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \tanh ^{-1}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\& \right ][x+c_1]\right \}\right \}\]
Maple ✓
cpu = 2.268 (sec), leaf count = 122
\[\left [x -\sqrt {a^{2}-y \left (x \right )^{2}}+\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-y \left (x \right )^{2}}}{y \left (x \right )}\right )}{\sqrt {a^{2}}}-\textit {\_C1} = 0, x +\sqrt {a^{2}-y \left (x \right )^{2}}-\frac {a^{2} \ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-y \left (x \right )^{2}}}{y \left (x \right )}\right )}{\sqrt {a^{2}}}-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[(a^2 - y[x]^2)*y'[x]^2 == y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-(a*ArcTanh[Sqrt[a^2 - #1^2]/a]) + Sqrt[a^2 - #1^2] & ][-x + C[1]]}, {y[x] -> InverseFunction[-(a*ArcTanh[Sqrt[a^2 - #1^2]/a]) + Sqrt[a^2 - #1^2] & ][x + C[1]]}}
Maple raw input
dsolve((a^2-y(x)^2)*diff(y(x),x)^2 = y(x)^2, y(x))
Maple raw output
[x-(a^2-y(x)^2)^(1/2)+a^2/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-y(x)^2)^(1/2))/y(x))-_C1 = 0, x+(a^2-y(x)^2)^(1/2)-a^2/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-y(x)^2)^(1/2))/y(x))-_C1 = 0]