ODE
\[ \left (1-x^2\right ) y'(x)=x y(x) (a y(x)+1) \] ODE Classification
[_separable]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.333856 (sec), leaf count = 31
\[\left \{\left \{y(x)\to -\frac {e^{c_1}}{-\sqrt {x^2-1}+a e^{c_1}}\right \}\right \}\]
Maple ✓
cpu = 0.021 (sec), leaf count = 22
\[\left [y \left (x \right ) = \frac {1}{\sqrt {x -1}\, \sqrt {1+x}\, \textit {\_C1} -a}\right ]\] Mathematica raw input
DSolve[(1 - x^2)*y'[x] == x*y[x]*(1 + a*y[x]),y[x],x]
Mathematica raw output
{{y[x] -> -(E^C[1]/(a*E^C[1] - Sqrt[-1 + x^2]))}}
Maple raw input
dsolve((-x^2+1)*diff(y(x),x) = x*y(x)*(1+a*y(x)), y(x))
Maple raw output
[y(x) = 1/((x-1)^(1/2)*(1+x)^(1/2)*_C1-a)]