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The differential equation

$\frac{dy}{dx}=\frac{48}{y^{\frac{1}{9}}+36x^2{y}^{\frac{1}{9}}}$

has an implicit general solution of the form $F(x,y)=K,$ where $K$ is an arbitrary constant.

In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form

$F(x,y)=G(x)+H(y)=K$

Find such a solution and then give the related functions requested.

$F(x,y)=G(x)+H(y)=\frac{9}{8}{y}^{\frac{8}{9}}+8arctan(6x)$

Example: Solve the differential equation implicitly in the form $F(x,y)=K$:$,y^{\text{'}}=\frac{27}{(y{)}^{\frac{1}{6}}+81x^2(y{)}^{\frac{1}{6}}}$

First rewrite the equation:

$y^{\text{'}}=\frac{27}{(y{)}^{\frac{1}{3}}(1+81x^2)}$

Integrate both sides:

$(y{)}^{\frac{1}{8}}dy=\frac{27}{?}dx$

using substitution $u=9x$ and $du=9dx{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{27}{(1+81x^2)}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{3}{(1+u^2)}du$