4.7.3 \(\left (1-x^2\right ) y'(x)=x y(x) (a y(x)+1)\)

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 = 0.0350801 (sec), leaf count = 33

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

Maple
cpu = 0.011 (sec), leaf count = 22

\[ \left \{ -\sqrt {-1+x}\sqrt {1+x}{\it \_C1}+ \left (y \relax (x ) \right ) ^{-1}+a=0 \right \} \] 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),'implicit')

Maple raw output

-(-1+x)^(1/2)*(1+x)^(1/2)*_C1+1/y(x)+a = 0