4.20.25 $$-a^2+y(x)^2 y'(x)^2+y(x)^2=0$$

ODE
$-a^2+y(x)^2 y'(x)^2+y(x)^2=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $$y$$

Mathematica
cpu = 0.209947 (sec), leaf count = 89

$\left \{\left \{y(x)\to -\sqrt {a^2-(x+c_1){}^2}\right \},\left \{y(x)\to \sqrt {a^2-(x+c_1){}^2}\right \},\left \{y(x)\to -\sqrt {a^2-(x-c_1){}^2}\right \},\left \{y(x)\to \sqrt {a^2-(x-c_1){}^2}\right \}\right \}$

Maple
cpu = 1.288 (sec), leaf count = 59

$\left [y \left (x \right ) = a, y \left (x \right ) = -a, y \left (x \right ) = \sqrt {-\textit {\_C1}^{2}+2 x \textit {\_C1} +a^{2}-x^{2}}, y \left (x \right ) = -\sqrt {-\textit {\_C1}^{2}+2 x \textit {\_C1} +a^{2}-x^{2}}\right ]$ Mathematica raw input

DSolve[-a^2 + y[x]^2 + y[x]^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[a^2 - (x + C[1])^2]}, {y[x] -> Sqrt[a^2 - (x + C[1])^2]}, {y[x]
-> -Sqrt[a^2 - (x - C[1])^2]}, {y[x] -> Sqrt[a^2 - (x - C[1])^2]}}

Maple raw input

dsolve(y(x)^2*diff(y(x),x)^2-a^2+y(x)^2 = 0, y(x))

Maple raw output

[y(x) = a, y(x) = -a, y(x) = (-_C1^2+2*_C1*x+a^2-x^2)^(1/2), y(x) = -(-_C1^2+2*_
C1*x+a^2-x^2)^(1/2)]