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

The solution for d/(dy)(x(y)^{2/3})+y^{2/3}=a^{2/3} is x(y)=a^{2/3}y^{1/3}-(3y)/5+(c_{1})/(y^{2/3)}

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d/(dy)(x(y)^{2/3})+y^{2/3}=a^{2/3}

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

x (y) = a^{\frac{2}{3}} y^{\frac{1}{3}} - \frac{3 y}{5} + \frac{c _{1}}{y ^{\frac{2}{3}}}

Solution steps

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

Isolate

(x (y) y^{\frac{2}{3}})^{'} : (x (y) y^{\frac{2}{3}})^{'} = a^{\frac{2}{3}} - y^{\frac{2}{3}}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

x (y) y^{\frac{2}{3}} = \int a^{\frac{2}{3}} - y^{\frac{2}{3}} d y

\int a^{\frac{2}{3}} - y^{\frac{2}{3}} d y = a^{\frac{2}{3}} y - \frac{3}{5} y^{\frac{5}{3}} + c_{1}

x (y) y^{\frac{2}{3}} = a^{\frac{2}{3}} y - \frac{3}{5} y^{\frac{5}{3}} + c_{1}

Isolate

x (y) : x (y) = a^{\frac{2}{3}} y^{\frac{1}{3}} - \frac{3 y}{5} + \frac{c _{1}}{y ^{\frac{2}{3}}}

x (y) = a^{\frac{2}{3}} y^{\frac{1}{3}} - \frac{3 y}{5} + \frac{c _{1}}{y ^{\frac{2}{3}}}

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

y^{^{'}} = x^{2} y^{- 3}

(x^{2} + 1) y^{^{'}} + 4 x y = 3 x

y^{^{'}} = - \frac{x - 7}{y - 3}, y (8) = 0

\frac{d y}{d t} = - 2 k (y^{2} + 2)

What is the solution for d/(dy)(x(y)^{2/3})+y^{2/3}=a^{2/3} ?
The solution for d/(dy)(x(y)^{2/3})+y^{2/3}=a^{2/3} is x(y)=a^{2/3}y^{1/3}-(3y)/5+(c_{1})/(y^{2/3)}