ODE No. 105

3.105 ODE No. 105

$x \frac{d}{d x} y (x) + a x {(y (x))}^{2} + b y (x) + c x + d = 0$

Mathematica: cpu = 0.187024 (sec), leaf count = 473 ${{y (x) \to \frac{c_{1} (i \sqrt{a} e^{- i \sqrt{a} \sqrt{c} x} (b (- \sqrt{c}) - i \sqrt{a} d) U (1 - \frac{- \sqrt{c} b - i \sqrt{a} d}{2 \sqrt{c}}, b + 1, 2 i \sqrt{a} \sqrt{c} x) - i \sqrt{a} \sqrt{c} e^{- i \sqrt{a} \sqrt{c} x} U (- \frac{- \sqrt{c} b - i \sqrt{a} d}{2 \sqrt{c}}, b, 2 i \sqrt{a} \sqrt{c} x)) - i \sqrt{a} \sqrt{c} e^{- i \sqrt{a} \sqrt{c} x} L_{\frac{b (- \sqrt{c}) - i \sqrt{a} d}{2 \sqrt{c}}}^{b - 1} (2 i \sqrt{a} \sqrt{c} x) - 2 i \sqrt{a} \sqrt{c} e^{- i \sqrt{a} \sqrt{c} x} L_{- 1 + \frac{b (- \sqrt{c}) - i \sqrt{a} d}{2 \sqrt{c}}}^{b} (2 i \sqrt{a} \sqrt{c} x)}{a (c_{1} e^{- i \sqrt{a} \sqrt{c} x} U (- \frac{- \sqrt{c} b - i \sqrt{a} d}{2 \sqrt{c}}, b, 2 i \sqrt{a} \sqrt{c} x) + e^{- i \sqrt{a} \sqrt{c} x} L_{\frac{b (- \sqrt{c}) - i \sqrt{a} d}{2 \sqrt{c}}}^{b - 1} (2 i \sqrt{a} \sqrt{c} x))}}}$

Maple: cpu = 0.202 (sec), leaf count = 844 ${y (x) = - 4 c^{2} (- 1 / 4_C 1 (a^{3} c^{2} d^{2} + a^{2} b^{2} c^{3} - 2 {(- a c)}^{3 / 2} a b c d - 2 {(- a c)}^{5 / 2} b d) U (1 / 2 \frac{{(- a c)}^{3 / 2} d + (2 \sqrt{- a c} d + c (b + 2)) a c}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) + (a c^{3} (a d - b \sqrt{- a c}) M (1 / 2 \frac{{(- a c)}^{3 / 2} d + (2 \sqrt{- a c} d + c (b + 2)) a c}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) + a c^{3} (b \sqrt{- a c} + a d) M (1 / 2 \frac{{(- a c)}^{3 / 2} d + 2 a d c \sqrt{- a c} + a b c^{2}}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) - 1 / 2 U (1 / 2 \frac{{(- a c)}^{3 / 2} d + 2 a d c \sqrt{- a c} + a b c^{2}}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c})_C 1 (b c - \sqrt{- a c} d)) a^{2} c^{2}) {(-_C 1 (a^{2} b^{2} c^{4} \sqrt{- a c} + 2 a c d^{2} {(- a c)}^{5 / 2} + d^{2} {(- a c)}^{7 / 2}) U (1 / 2 \frac{{(- a c)}^{3 / 2} d + (2 \sqrt{- a c} d + c (b + 2)) a c}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) + 4 a^{2} c^{4} (a^{2} c^{2} (\sqrt{- a c} d + b c) M (1 / 2 \frac{{(- a c)}^{3 / 2} d + (2 \sqrt{- a c} d + c (b + 2)) a c}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) + a^{2} c^{2} (b c - \sqrt{- a c} d) M (1 / 2 \frac{{(- a c)}^{3 / 2} d + 2 a d c \sqrt{- a c} + a b c^{2}}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c}) + 1 / 2 U (1 / 2 \frac{{(- a c)}^{3 / 2} d + 2 a d c \sqrt{- a c} + a b c^{2}}{c^{2} a}, \frac{{(- a c)}^{3 / 2} d + (\sqrt{- a c} d + c (b + 1)) a c}{c^{2} a}, 2 x \sqrt{- a c})_C 1 (b \sqrt{- a c} + a d)))}^{- 1}}$

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