4.9.4 \(\sqrt {(1-x) x (1-a x)} y'(x)=\sqrt {(1-y(x)) y(x) (1-a y(x))}\)

ODE
\[ \sqrt {(1-x) x (1-a x)} y'(x)=\sqrt {(1-y(x)) y(x) (1-a y(x))} \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.317921 (sec), leaf count = 100

\[\left \{\left \{y(x)\to \text {ns}\left (\frac {1}{2} i \sqrt {a} c_1-F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2 \left (-1+\text {sn}\left (\frac {1}{2} i \sqrt {a} c_1-F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2\right )\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 38

\[ \left \{ \int \!{\frac {1}{\sqrt {x \left (-1+x \right ) \left (ax-1 \right ) }}}\,{\rm d}x-\int ^{y \relax (x ) }\!{\frac {1}{\sqrt {{\it \_a}\, \left ({\it \_a}-1 \right ) \left ({\it \_a}\,a-1 \right ) }}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[Sqrt[(1 - x)*x*(1 - a*x)]*y'[x] == Sqrt[(1 - y[x])*y[x]*(1 - a*y[x])],y[x],x]

Mathematica raw output

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

Maple raw input

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

Maple raw output

Int(1/(x*(-1+x)*(a*x-1))^(1/2),x)-Intat(1/(_a*(_a-1)*(_a*a-1))^(1/2),_a = y(x))+
_C1 = 0