(dy)/(dx)=8x(6y+5)^{1/2}

Lösung

\frac{d y}{d x} = 8 x (6 y + 5)^{\frac{1}{2}}

y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} + 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}, y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} - 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}

Schritte zur Lösung

\frac{d y}{d x} = 8 x (6 y + 5)^{\frac{1}{2}}

Solve separable ODE:

y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} + 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}, y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} - 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}

y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} + 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}, y = \frac{- 10 + 9 c _{1}^{2} + 72 c _{1} x ^{2} + 144 x ^{4} - 9 ( 256 x ^{8} + 256 c _{1} x ^{6} + 96 c _{1}^{2} x ^{4} + 16 c _{1}^{3} x ^{2} + c _{1}^{4} ) ^{\frac{1}{2}}}{12}

\int \frac{- 38 y + 25}{- y ( 7 y - 5 )} d y

\int y + z d x

\frac{d ^{4}}{d x ^{4}} (18 e^{x^{2}})

x \to \infty lim (2^{x} - 1)

\frac{d}{d x} (4 e^{- x^{4}})