** Final answer to the problem

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** Step-by-step Solution **

** How should I solve this problem?

- Separable Differential Equation
- Exact Differential Equation
- Linear Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
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We identify that the differential equation $\frac{dy}{dx}+\frac{-y}{x}=\frac{x}{3y}$ is a Bernoulli differential equation since it's of the form $\frac{dy}{dx}+P(x)y=Q(x)y^n$, where $n$ is any real number different from $0$ and $1$. To solve this equation, we can apply the following substitution. Let's define a new variable $u$ and set it equal to

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Plug in the value of $n$, which equals $-1$

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Simplify

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Isolate the dependent variable $y$

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Differentiate both sides of the equation with respect to the independent variable $x$

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Now, substitute $\frac{dy}{dx}=\frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}$ and $y=\sqrt{u}$ on the original differential equation

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Simplify

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We need to cancel the term that is in front of $\frac{du}{dx}$. We can do that by multiplying the whole differential equation by $\frac{1}{2}\sqrt{u}$

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Multiply both sides by $\frac{1}{2}\sqrt{u}$

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Expand and simplify. Now we see that the differential equation looks like a linear differential equation, because we removed the original $y^{-1}$ term

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Divide all the terms of the differential equation by $\frac{1}{4}$

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Simplifying

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We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=\frac{-1}{\frac{1}{2}x}$ and $Q(x)=\frac{2}{3}x$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

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To find $\mu(x)$, we first need to calculate $\int P(x)dx$

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So the integrating factor $\mu(x)$ is

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Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify

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We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$

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Integrate both sides of the differential equation with respect to $dx$

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Simplify the left side of the differential equation

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Solve the integral $\int\frac{2}{3}x^{-1}dx$ and replace the result in the differential equation

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Replace $u$ with the value $y^{2}$

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Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

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Multiply the fraction and term

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

** Final answer to the problem

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