## Final Answer

## Step-by-step Solution

Problem to solve:

Take $\frac{2}{3}$ out of the fraction

Simplify the fraction $\frac{\frac{2}{3}x}{y^2}$

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$

Simplify the fraction $\frac{3}{2}\left(\frac{y^{3}}{3}\right)$

Solve the integral $\int\frac{3}{2}y^2dy$ and replace the result in the differential equation

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

Solve the integral $\int xdx$ and replace the result in the differential equation

Cancel $\frac{1}{2}$ from both sides of the equation

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

Removing the variable's exponent

Find the explicit solution to the differential equation