ODE No. 144

3.144 ODE No. 144

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

Mathematica: cpu = 0.168521 (sec), leaf count = 1712 ${{y (x) \to \frac{a^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}} α^{- \frac{i \sqrt{4 a c - 1} α + α}{α^{2}} + \frac{i \sqrt{4 a c - 1}}{α} + 1} b^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}} (\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}) {(x^{α})}^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α} - 1} J_{\frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}}} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) Γ (\frac{\sqrt{1 - 4 a c}}{α} + 1) x^{α - 1} + \frac{1}{2} a^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α} + \frac{1}{2}} α^{\frac{i \sqrt{4 a c - 1}}{α} - \frac{i \sqrt{4 a c - 1} α + α}{α^{2}}} b^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α} + \frac{1}{2}} {(x^{α})}^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α} - \frac{1}{2}} (J_{\frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}} - 1} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) - J_{\frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}} + 1} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α})) Γ (\frac{\sqrt{1 - 4 a c}}{α} + 1) x^{α - 1} + c_{1} (a^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}} α^{- \frac{α - i α \sqrt{4 a c - 1}}{α^{2}} - \frac{i \sqrt{4 a c - 1}}{α} + 1} b^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}} (\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}) {(x^{α})}^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α} - 1} J_{- \frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}}} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) Γ (1 - \frac{\sqrt{1 - 4 a c}}{α}) x^{α - 1} + \frac{1}{2} a^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α} + \frac{1}{2}} α^{- \frac{α - i α \sqrt{4 a c - 1}}{α^{2}} - \frac{i \sqrt{4 a c - 1}}{α}} b^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α} + \frac{1}{2}} {(x^{α})}^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α} - \frac{1}{2}} (J_{- \frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}} - 1} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) - J_{1 - \frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}}} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α})) Γ (1 - \frac{\sqrt{1 - 4 a c}}{α}) x^{α - 1})}{a (a^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}} b^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}} {(x^{α})}^{\frac{α - i α \sqrt{4 a c - 1}}{2 α^{2}} + \frac{i \sqrt{4 a c - 1}}{2 α}} J_{- \frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}}} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) c_{1} Γ (1 - \frac{\sqrt{1 - 4 a c}}{α}) α^{- \frac{α - i α \sqrt{4 a c - 1}}{α^{2}} - \frac{i \sqrt{4 a c - 1}}{α}} + a^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}} b^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}} {(x^{α})}^{\frac{i \sqrt{4 a c - 1} α + α}{2 α^{2}} - \frac{i \sqrt{4 a c - 1}}{2 α}} J_{\frac{\sqrt{α^{2} - 4 a α^{2} c}}{α^{2}}} (\frac{2 \sqrt{a} \sqrt{b} \sqrt{x^{α}}}{α}) Γ (\frac{\sqrt{1 - 4 a c}}{α} + 1) α^{\frac{i \sqrt{4 a c - 1}}{α} - \frac{i \sqrt{4 a c - 1} α + α}{α^{2}}})}}}$

Maple: cpu = 0.078 (sec), leaf count = 244 ${y (x) = - \frac{1}{2 a x} ((- \sqrt{- 4 a c + 1}_C 1 -_C 1) Y_{\frac{1}{α} \sqrt{- 4 a c + 1}} (2 \frac{\sqrt{a b} x^{α / 2}}{α}) + 2 x^{α / 2} Y_{\frac{\sqrt{- 4 a c + 1} + α}{α}} (2 \frac{\sqrt{a b} x^{α / 2}}{α}) \sqrt{a b}_C 1 + (- \sqrt{- 4 a c + 1} - 1) J_{\frac{1}{α} \sqrt{- 4 a c + 1}} (2 \frac{\sqrt{a b} x^{α / 2}}{α}) + 2 J_{\frac{\sqrt{- 4 a c + 1} + α}{α}} (2 \frac{\sqrt{a b} x^{α / 2}}{α}) \sqrt{a b} x^{α / 2}) {(Y_{\frac{1}{α} \sqrt{- 4 a c + 1}} (2 \frac{\sqrt{a b} x^{α / 2}}{α})_C 1 + J_{\frac{1}{α} \sqrt{- 4 a c + 1}} (2 \frac{\sqrt{a b} x^{α / 2}}{α}))}^{- 1}}$

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