ODE No. 501

3.501 ODE No. 501

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

Mathematica: cpu = 33.600767 (sec), leaf count = 913 ${Solve [{y (x) = \frac{b K $ 1419264 - \sqrt{- {K $ 1419264}^{2} (- b^{2} + 4 a {K $ 1419264}^{2} x b + 4 d x b + 4 a c {K $ 1419264}^{2} + 4 c d)}}{2 (a {K $ 1419264}^{2} + d)}, x = \frac{- b^{2} c_{1}^{2} d^{4} - a b^{2} {K $ 1419264}^{2} c_{1}^{2} d^{3} + 2 b^{2} c_{1} \log (K $ 1419264) d^{5 / 2} - 2 b^{2} c_{1} \log (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) d^{5 / 2} - 4 c d^{2} + 2 b^{2} \sqrt{a {K $ 1419264}^{2} + d} c_{1} d^{2} + 2 a b^{2} {K $ 1419264}^{2} c_{1} \log (K $ 1419264) d^{3 / 2} - 2 a b^{2} {K $ 1419264}^{2} c_{1} \log (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) d^{3 / 2} - 4 a c {K $ 1419264}^{2} d - b^{2} \log^{2} (K $ 1419264) d - b^{2} \log^{2} (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) d + 2 b^{2} \log (K $ 1419264) \log (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) d - 2 b^{2} \sqrt{a {K $ 1419264}^{2} + d} \log (K $ 1419264) \sqrt{d} + 2 b^{2} \sqrt{a {K $ 1419264}^{2} + d} \log (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) \sqrt{d} - a b^{2} {K $ 1419264}^{2} \log^{2} (K $ 1419264) - a b^{2} {K $ 1419264}^{2} \log^{2} (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d}) + 2 a b^{2} {K $ 1419264}^{2} \log (K $ 1419264) \log (d + \sqrt{a {K $ 1419264}^{2} + d} \sqrt{d})}{4 b d (a {K $ 1419264}^{2} + d)}}, {y (x), K $ 1419264}], Solve [{y (x) = \frac{b K $ 1419269 + \sqrt{- {K $ 1419269}^{2} (- b^{2} + 4 a {K $ 1419269}^{2} x b + 4 d x b + 4 a c {K $ 1419269}^{2} + 4 c d)}}{2 (a {K $ 1419269}^{2} + d)}, x = \frac{- b^{2} c_{1}^{2} d^{4} - a b^{2} {K $ 1419269}^{2} c_{1}^{2} d^{3} + 2 b^{2} c_{1} \log (K $ 1419269) d^{5 / 2} - 2 b^{2} c_{1} \log (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) d^{5 / 2} - 4 c d^{2} + 2 b^{2} \sqrt{a {K $ 1419269}^{2} + d} c_{1} d^{2} + 2 a b^{2} {K $ 1419269}^{2} c_{1} \log (K $ 1419269) d^{3 / 2} - 2 a b^{2} {K $ 1419269}^{2} c_{1} \log (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) d^{3 / 2} - 4 a c {K $ 1419269}^{2} d - b^{2} \log^{2} (K $ 1419269) d - b^{2} \log^{2} (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) d + 2 b^{2} \log (K $ 1419269) \log (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) d - 2 b^{2} \sqrt{a {K $ 1419269}^{2} + d} \log (K $ 1419269) \sqrt{d} + 2 b^{2} \sqrt{a {K $ 1419269}^{2} + d} \log (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) \sqrt{d} - a b^{2} {K $ 1419269}^{2} \log^{2} (K $ 1419269) - a b^{2} {K $ 1419269}^{2} \log^{2} (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d}) + 2 a b^{2} {K $ 1419269}^{2} \log (K $ 1419269) \log (d + \sqrt{a {K $ 1419269}^{2} + d} \sqrt{d})}{4 b d (a {K $ 1419269}^{2} + d)}}, {y (x), K $ 1419269}]}$

Maple: cpu = 4.259 (sec), leaf count = 287 ${[x (_T) = - \frac{1}{4 b d} ({(\ln (2))}^{2} \sqrt{{_T}^{2} a + d} b^{2} + 2 \ln (2) \ln (\frac{\sqrt{d} \sqrt{{_T}^{2} a + d} + d}{_T}) \sqrt{{_T}^{2} a + d} b^{2} + 4 \ln (2) \sqrt{d} \sqrt{{_T}^{2} a + d}_C 1 b + {(\ln (\frac{1}{_T} (\sqrt{d} \sqrt{{_T}^{2} a + d} + d)))}^{2} \sqrt{{_T}^{2} a + d} b^{2} + 4 \ln (\frac{\sqrt{d} \sqrt{{_T}^{2} a + d} + d}{_T}) \sqrt{d} \sqrt{{_T}^{2} a + d}_C 1 b + 4 d \sqrt{{_T}^{2} a + d} {_C 1}^{2} - 2 \ln (2) \sqrt{d} b^{2} - 2 \ln (\frac{\sqrt{d} \sqrt{{_T}^{2} a + d} + d}{_T}) \sqrt{d} b^{2} + 4 c d \sqrt{{_T}^{2} a + d} - 4 d_C 1 b) \frac{1}{\sqrt{{_T}^{2} a + d}}, y (_T) = \frac{_T}{2} (b \ln (2) + b \ln (\frac{1}{_T} (\sqrt{d} \sqrt{{_T}^{2} a + d} + d)) + 2_C 1 \sqrt{d}) \frac{1}{\sqrt{d}} \frac{1}{\sqrt{{_T}^{2} a + d}}]}$

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