** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

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

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The differential equation $3y^2dy-2xdx=0$ 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 $f(x,y)=C$

**Step 3**. Or become premium for the price of a latte.

The integral of a function times a constant ($-2$) is equal to the constant times 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

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

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

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

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Now take the partial derivative of $-x^2$ with respect to $y$ to get

**Step 6**. Or become premium for the price of a latte.

Integrate both sides with respect to $y$

The integral of a function times a constant ($3$) is equal to the constant times 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 $2$

Any expression multiplied by $1$ is equal to itself

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Find $g(y)$ integrating both sides

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

**Step 9**. Or become premium for the price of a latte.

We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $-x^2$ from both sides of the equation

Removing the variable's exponent raising both sides of the equation to the power $$

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Find the explicit solution to the differential equation

** Final Answer

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