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

[_quadrature]

Book solution method
Change of variable

Mathematica
cpu = 0.091701 (sec), leaf count = 71

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

Maple
cpu = 0.143 (sec), leaf count = 50

\[ \left \{ x-{\frac {\arcsin \left (\ln \left (y \relax (x ) \right ) \right ) }{a}}-{\it \_C1}=0,x+{\frac {\arcsin \left (\ln \left (y \relax (x ) \right ) \right ) }{a}}-{\it \_C1}=0,y \relax (x ) ={{\rm e}^{{\it RootOf} \left ({a}^{2} \left ({{\rm e}^{{\it \_Z}}} \right ) ^{2} \left ({{\it \_Z}}^{2}-1 \right ) \right ) }} \right \} \] 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^((E^(I*a*x - C[1]) + E^((-I)*a*x + C[1]))/2)}, {y[x] -> E^((E^((-I)*
a*x - C[1]) + E^(I*a*x + C[1]))/2)}}

Maple raw input

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

Maple raw output

y(x) = exp(RootOf(a^2*exp(_Z)^2*(_Z^2-1))), x-1/a*arcsin(ln(y(x)))-_C1 = 0, x+1/
a*arcsin(ln(y(x)))-_C1 = 0