4.7.38 \(2 (1-x) x y'(x)+(1-x) y(x)^2+x=0\)

ODE
\[ 2 (1-x) x y'(x)+(1-x) y(x)^2+x=0 \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 0.328114 (sec), leaf count = 63

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

Maple ✓
cpu = 0.162 (sec), leaf count = 97

\[\left [y \left (x \right ) = \frac {x \left (\LegendreQ \left (-\frac {1}{2}, 1, \frac {2-x}{x}\right ) \textit {\_C1} -\LegendreQ \left (\frac {1}{2}, 1, \frac {2-x}{x}\right ) \textit {\_C1} +\LegendreP \left (-\frac {1}{2}, 1, \frac {2-x}{x}\right )-\LegendreP \left (\frac {1}{2}, 1, \frac {2-x}{x}\right )\right )}{2 \left (\LegendreQ \left (-\frac {1}{2}, 1, \frac {2-x}{x}\right ) \textit {\_C1} +\LegendreP \left (-\frac {1}{2}, 1, \frac {2-x}{x}\right )\right ) \left (x -1\right )}\right ]\] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = 1/2*x*(LegendreQ(-1/2,1,(2-x)/x)*_C1-LegendreQ(1/2,1,(2-x)/x)*_C1+Legend
reP(-1/2,1,(2-x)/x)-LegendreP(1/2,1,(2-x)/x))/(LegendreQ(-1/2,1,(2-x)/x)*_C1+Leg
endreP(-1/2,1,(2-x)/x))/(x-1)]