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 = 0.119322 (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 = 0.553 (sec), leaf count = 30

\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C1}\,{\it EllipticCE} \left ( x \right ) +{\it EllipticE} \left ( x \right ) -{\it EllipticK} \left ( x \right ) }{{\it \_C1}\,{\it EllipticCE} \left ( x \right ) -{\it \_C1}\,{\it EllipticCK} \left ( x \right ) +{\it EllipticE} \left ( x \right ) }} \right \} \] 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),'implicit')

Maple raw output

y(x) = (_C1*EllipticCE(x)+EllipticE(x)-EllipticK(x))/(_C1*EllipticCE(x)-_C1*Elli
pticCK(x)+EllipticE(x))