What is the solution for (dy)/(dx)=(xy)^{1/3},y(2)=1 ?

The solution for (dy)/(dx)=(xy)^{1/3},y(2)=1 is

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(dy)/(dx)=(xy)^{1/3},y(2)=1

\frac{d y}{d x} = (x y)^{\frac{1}{3}}, y (2) = 1

y = \frac{2 ^{\frac{2}{3}} x ^{4} + 6 x ^{\frac{8}{3}} ( 2 ^{\frac{2}{3}} - 2 ) + 12 x ^{\frac{4}{3}} ( 2 ^{\frac{2}{3}} + 2 \cdot 2 ^{\frac{1}{3}} - 4 ) + 2 ^{\frac{4}{3}} \cdot 24 - 2 ^{\frac{2}{3}} \cdot 8 - 48}{2 \cdot 2 ^{\frac{5}{6}}}

Solution steps

\frac{d y}{d x} = (x y)^{\frac{1}{3}}

Solve separable ODE:

y = \frac{2 ^{\frac{2}{3}} x ^{4} + 6 x ^{\frac{8}{3}} ( 2 ^{\frac{2}{3}} - 2 ) + 12 x ^{\frac{4}{3}} ( 2 ^{\frac{2}{3}} + 2 \cdot 2 ^{\frac{1}{3}} - 4 ) + 2 ^{\frac{4}{3}} \cdot 24 - 2 ^{\frac{2}{3}} \cdot 8 - 48}{2 \cdot 2 ^{\frac{5}{6}}}

y = \frac{2 ^{\frac{2}{3}} x ^{4} + 6 x ^{\frac{8}{3}} ( 2 ^{\frac{2}{3}} - 2 ) + 12 x ^{\frac{4}{3}} ( 2 ^{\frac{2}{3}} + 2 \cdot 2 ^{\frac{1}{3}} - 4 ) + 2 ^{\frac{4}{3}} \cdot 24 - 2 ^{\frac{2}{3}} \cdot 8 - 48}{2 \cdot 2 ^{\frac{5}{6}}}

\frac{d}{d x} (7 x e^{- 9 x})

f (x) = - sin (2 x)

x \to 0 lim (\frac{3 x ^{2}}{1 - cos ^{2} ( \frac{x}{2} )})

\int \frac{x ^{3} + x ^{2} + 2}{x ^{2} - x} d x

\int \frac{1}{( x ^{2} + 2 x + 3 ) ^{\frac{3}{2}}} d x

What is the solution for (dy)/(dx)=(xy)^{1/3},y(2)=1 ?
The solution for (dy)/(dx)=(xy)^{1/3},y(2)=1 is