ODE No. 64

2.64 ODE No. 64

$y^{'} (x) - \sqrt{\frac{a y (x)^{2} + b y (x) + c}{a x^{2} + b x + c}} = 0$ ✓ Mathematica : cpu = 0.174964 (sec), leaf count = 269

${{y (x) \to \frac{e^{- \sqrt{a} c_{1}} (8 a^{3 / 2} c e^{2 \sqrt{a} c_{1}} \sqrt{a x^{2} + b x + c} - 8 a^{3 / 2} c \sqrt{a x^{2} + b x + c} + 8 a^{2} c x e^{2 \sqrt{a} c_{1}} + 8 a^{2} c x + 2 b^{3} e^{\sqrt{a} c_{1}} - b^{3} e^{2 \sqrt{a} c_{1}} - 2 \sqrt{a} b^{2} e^{2 \sqrt{a} c_{1}} \sqrt{a x^{2} + b x + c} + 2 \sqrt{a} b^{2} \sqrt{a x^{2} + b x + c} - 2 a b^{2} x e^{2 \sqrt{a} c_{1}} - 2 a b^{2} x - 8 a b c e^{\sqrt{a} c_{1}} + 4 a b c e^{2 \sqrt{a} c_{1}} + 4 a b c - b^{3})}{a (16 a c - 4 b^{2})}}}$

✓ Maple : cpu = 0.084 (sec), leaf count = 124

${- 1 \sqrt{\frac{a {(y (x))}^{2} + b y (x) + c}{a x^{2} + b x + c}} \sqrt{a x^{2} + b x + c} \ln (\frac{1}{2} (2 \sqrt{a x^{2} + b x + c} \sqrt{a} + 2 a x + b) \frac{1}{\sqrt{a}}) \frac{1}{\sqrt{a {(y (x))}^{2} + b y (x) + c}} \frac{1}{\sqrt{a}} + 1 \ln (1 (a y (x) + \frac{b}{2}) \frac{1}{\sqrt{a}} + \sqrt{a {(y (x))}^{2} + b y (x) + c}) \frac{1}{\sqrt{a}} +_C 1 = 0}$

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