##### 4.8.16 $$\left (1-x^2\right ) x y'(x)+\left (1-x^2\right ) y(x)^2+x^2=0$$

ODE
$\left (1-x^2\right ) x y'(x)+\left (1-x^2\right ) y(x)^2+x^2=0$ ODE Classiﬁcation

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.348736 (sec), leaf count = 73

$\left \{\left \{y(x)\to -\frac {2 \left (\pi G_{2,2}^{2,0}\left (x^2|\begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,1 \\\end {array}\right )+c_1 \left (K\left (x^2\right )-E\left (x^2\right )\right )\right )}{\pi G_{2,2}^{2,0}\left (x^2|\begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,0 \\\end {array}\right )+2 c_1 E\left (x^2\right )}\right \}\right \}$

Maple
cpu = 0.101 (sec), leaf count = 45

$\left [y \left (x \right ) = -\frac {\EllipticK \left (x \right )}{\textit {\_C1} \EllipticCE \left (x \right )-\textit {\_C1} \EllipticCK \left (x \right )+\EllipticE \left (x \right )}+\frac {\textit {\_C1} \EllipticCE \left (x \right )+\EllipticE \left (x \right )}{\textit {\_C1} \EllipticCE \left (x \right )-\textit {\_C1} \EllipticCK \left (x \right )+\EllipticE \left (x \right )}\right ]$ Mathematica raw input

DSolve[x^2 + (1 - x^2)*y[x]^2 + x*(1 - x^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-2*(C[1]*(-EllipticE[x^2] + EllipticK[x^2]) + Pi*MeijerG[{{}, {1/2, 3
/2}}, {{0, 1}, {}}, x^2]))/(2*C[1]*EllipticE[x^2] + Pi*MeijerG[{{}, {1/2, 3/2}},
 {{0, 0}, {}}, x^2])}}

Maple raw input

dsolve(x*(-x^2+1)*diff(y(x),x)+x^2+(-x^2+1)*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = -1/(_C1*EllipticCE(x)-_C1*EllipticCK(x)+EllipticE(x))*EllipticK(x)+(_C1*
EllipticCE(x)+EllipticE(x))/(_C1*EllipticCE(x)-_C1*EllipticCK(x)+EllipticE(x))]