## Final Answer

## Step-by-step Solution

Problem to solve:

Choose the solving method

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

Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$

The differential equation $y^2dy-\frac{2}{3}xdx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$

Find the derivative of $M(x,y)$ with respect to $y$

The derivative of the constant function ($-\frac{2}{3}x$) is equal to zero

Find the derivative of $N(x,y)$ with respect to $x$

The derivative of the constant function ($y^2$) is equal to zero

Using the test for exactness, we check that the differential equation is exact

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

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 $n=1$

Since $y$ is treated as a constant, we add a function of $y$ as constant of integration

Integrate $M(x,y)$ with respect to $x$ to get

The derivative of the constant function ($-\frac{1}{3}x^2$) is equal to zero

The derivative of $g(y)$ is $g'(y)$

Now take the partial derivative of $-\frac{1}{3}x^2$ with respect to $y$ to get

Simplify and isolate $g'(y)$

$x+0=x$, where $x$ is any expression

Rearrange the equation

Set $y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$

Integrate both sides with respect to $y$

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 $2$

Find $g(y)$ integrating both sides

We have found our $f(x,y)$ and it equals

Then, the solution to the differential equation is

We need to isolate the dependent variable $y$, we can do that by subtracting $-\frac{1}{3}x^2$ from both sides of the equation

Simplify the fraction

Eliminate the $\frac{1}{3}$ from the left, multiplying both sides of the equation by the inverse of $\frac{1}{3}$

Solve the product $3\left(C_0+\frac{1}{3}x^2\right)$

We can rename $3C_0$ as other constant

Removing the variable's exponent

Find the explicit solution to the differential equation