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

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

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(dy)/(dx)y=(1/3)(x^2+2)^{3/2}

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

y = \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}, y = - \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}

Solution steps

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

Solve separable ODE:

y = \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}, y = - \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}

y = \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}, y = - \frac{3 x ( x ^{2} + 2 ) ^{\frac{1}{2}} + x ( x ^{2} + 2 ) ^{\frac{3}{2}} + 6 ln ( \frac{x}{2 ^{\frac{1}{2}}} + \frac{x ^{2} + 2}{2} ) + c _{1}}{6}

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

\frac{d y}{d x} = \frac{( y ln ( x ) )}{x}

(x^{2} + 25) \frac{d y}{d x} + x y = 9 x

(x + 1) d y - 3 y d x = 0

y^{^{'}} - \frac{x sin ( x )}{3 y ^{2}} = 0, y (0) = 2

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