##### 4.16.18 $$y'(x)^2=a^2 y(x)^2 \left (1-\log ^2(y(x))\right )$$

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

Book solution method
Change of variable

Mathematica
cpu = 0.462 (sec), leaf count = 185

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

Maple
cpu = 0.315 (sec), leaf count = 47

$[y \left (x \right ) = {\mathrm e}^{\RootOf \left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )}, y \left (x \right ) = {\mathrm e}^{-\sin \left (\textit {\_C1} a -a x \right )}, y \left (x \right ) = {\mathrm e}^{\sin \left (\textit {\_C1} a -a x \right )}]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = exp(RootOf(a^2*exp(_Z)^2*(_Z^2-1))), y(x) = 1/exp(sin(_C1*a-a*x)), y(x)
= exp(sin(_C1*a-a*x))]