ODE No. 1810

8.220 ODE No. 1810

$\sqrt{y (x)} \frac{d^{2}}{d x^{2}} y (x) - a = 0$

Mathematica: cpu = 0.060008 (sec), leaf count = 1677 ${{y (x) \to \frac{3 c_{1}^{2}}{16 a^{2}} + \frac{\sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}{768 \sqrt[3]{2}} - \frac{- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1}}{384 2^{2 / 3} \sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}}, {y (x) \to \frac{3 c_{1}^{2}}{16 a^{2}} - \frac{(1 - i \sqrt{3}) \sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}{1536 \sqrt[3]{2}} + \frac{(1 + i \sqrt{3}) (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})}{768 2^{2 / 3} \sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}}, {y (x) \to \frac{3 c_{1}^{2}}{16 a^{2}} - \frac{(1 + i \sqrt{3}) \sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}{1536 \sqrt[3]{2}} + \frac{(1 - i \sqrt{3}) (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})}{768 2^{2 / 3} \sqrt[3]{- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2} + \sqrt{4 (- \frac{2304 c_{1}^{4}}{a^{4}} - 663552 x^{2} c_{1} - 663552 c_{2}^{2} c_{1} - 1327104 x c_{2} c_{1})^{3} + (- \frac{221184 c_{1}^{6}}{a^{6}} + \frac{159252480 x^{2} c_{1}^{3}}{a^{2}} + \frac{159252480 c_{2}^{2} c_{1}^{3}}{a^{2}} + \frac{318504960 x c_{2} c_{1}^{3}}{a^{2}} + 2293235712 a^{2} x^{4} + 2293235712 a^{2} c_{2}^{4} + 9172942848 a^{2} x c_{2}^{3} + 13759414272 a^{2} x^{2} c_{2}^{2} + 9172942848 a^{2} x^{3} c_{2})^{2}}}}}}$

Maple: cpu = 0.109 (sec), leaf count = 55 ${\int^{y (x)} \frac{1}{\sqrt{4 \sqrt{_a} a -_C 1}} d_a - x -_C 2 = 0, \int^{y (x)} - \frac{1}{\sqrt{4 \sqrt{_a} a -_C 1}} d_a - x -_C 2 = 0}$

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