[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Use new variable
Mathematica ✓
cpu = 0.0914421 (sec), leaf count = 110
Maple ✓
cpu = 0.052 (sec), leaf count = 136
DSolve[(1 - a*y[x])*y'[x]^2 == a*y[x],y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[ArcSin[Sqrt[a]*Sqrt[#1]]/Sqrt[a] + Sqrt[#1]*Sqrt[1 - a
*#1] & ][-(Sqrt[a]*x) + C[1]]}, {y[x] -> InverseFunction[ArcSin[Sqrt[a]*Sqrt[#1]
]/Sqrt[a] + Sqrt[#1]*Sqrt[1 - a*#1] & ][Sqrt[a]*x + C[1]]}}
Maple raw input
dsolve((1-a*y(x))*diff(y(x),x)^2 = a*y(x), y(x),'implicit')
Maple raw output
y(x) = 0, x+1/a*(-a^2*y(x)^2+a*y(x))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*(y
(x)-1/2/a)/(-a^2*y(x)^2+a*y(x))^(1/2))-_C1 = 0, x-1/a*(-a^2*y(x)^2+a*y(x))^(1/2)
-1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*(y(x)-1/2/a)/(-a^2*y(x)^2+a*y(x))^(1/2))-_C1
= 0