** Final Answer

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** Step-by-step Solution **

** 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 $f(x,y)=C$

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

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

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

The derivative of the constant function ($3y^2$) is equal to zero

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

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

Simplify and isolate $g'(y)$

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

Rearrange the equation

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Set $3y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$

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

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Then, the solution to the differential equation is

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

Multiply $-1$ times $-1$

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{3}$

Divide $1$ by $3$

Simplify $\sqrt[3]{y^{3}}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $\frac{1}{3}$

Multiply $3$ times $\frac{1}{3}$

Multiply $3$ times $\frac{1}{3}$

Divide $1$ by $3$

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Find the explicit solution to the differential equation. We need to isolate the variable $y$

** Final Answer

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