2(1 −x)xy′(x) + (1 −x)y(x)2 + x = 0

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 = 0.0746079 (sec), leaf count = 63

${{y (x) \to - \frac{2 (π G_{2, 2}^{2, 0} (x | \begin{matrix} \frac{1}{2}, \frac{3}{2} \\ 0, 1 \end{matrix}) + c_{1} (K (x) - E (x)))}{π G_{2, 2}^{2, 0} (x | \begin{matrix} \frac{1}{2}, \frac{3}{2} \\ 0, 0 \end{matrix}) + 2 c_{1} E (x)}}}$

Maple ✓
cpu = 0.203 (sec), leaf count = 97

${y (x) = \frac{x}{- 2 + 2 x} (L e g e n d r e Q (- \frac{1}{2}, 1, \frac{2 - x}{x})_C 1 - L e g e n d r e Q (\frac{1}{2}, 1, \frac{2 - x}{x})_C 1 + L e g e n d r e P (- \frac{1}{2}, 1, \frac{2 - x}{x}) - L e g e n d r e P (\frac{1}{2}, 1, \frac{2 - x}{x})) {(L e g e n d r e Q (- \frac{1}{2}, 1, \frac{2 - x}{x})_C 1 + L e g e n d r e P (- \frac{1}{2}, 1, \frac{2 - x}{x}))}^{- 1}}$ 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),'implicit')

Maple raw output

y(x) = 1/2*x*(LegendreQ(-1/2,1,(2-x)/x)*_C1-LegendreQ(1/2,1,(2-x)/x)*_C1+Legendr
eP(-1/2,1,(2-x)/x)-LegendreP(1/2,1,(2-x)/x))/(LegendreQ(-1/2,1,(2-x)/x)*_C1+Lege
ndreP(-1/2,1,(2-x)/x))/(-1+x)

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4.7.38 2(1−x)xy′(x)+(1−x)y(x)2+x=0

4.7.38 $2 (1 - x) x y^{'} (x) + (1 - x) y (x)^{2} + x = 0$