##### 4.19.40 $$x^2 \left (a^2-x^2\right ) y'(x)^2+1=0$$

ODE
$x^2 \left (a^2-x^2\right ) y'(x)^2+1=0$ ODE Classiﬁcation

[_quadrature]

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

Mathematica
cpu = 0.172674 (sec), leaf count = 120

$\left \{\left \{y(x)\to -\frac {x \sqrt {x^2-a^2} \tan ^{-1}\left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1\right \},\left \{y(x)\to \frac {x \sqrt {x^2-a^2} \tan ^{-1}\left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1\right \}\right \}$

Maple
cpu = 0.089 (sec), leaf count = 90

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

DSolve[1 + x^2*(a^2 - x^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((x*Sqrt[-a^2 + x^2]*ArcTan[Sqrt[-a^2 + x^2]/a])/(a*Sqrt[-(a^2*x^2) +
 x^4])) + C[1]}, {y[x] -> (x*Sqrt[-a^2 + x^2]*ArcTan[Sqrt[-a^2 + x^2]/a])/(a*Sqr
t[-(a^2*x^2) + x^4]) + C[1]}}

Maple raw input

dsolve(x^2*(a^2-x^2)*diff(y(x),x)^2+1 = 0, y(x))

Maple raw output

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