** Final answer to the problem

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** Step-by-step Solution ** **

** How should I solve this problem?

- Linear Differential Equation
- Exact Differential Equation
- Separable 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
- Load more...

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

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Multiplying the fraction by $-1$

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Multiply both sides of the equation by $dx$

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The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

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Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete

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Combine and simplify all terms in the same fraction with common denominator $3yx$

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Divide both sides of the equation by $dx$

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We can identify that the differential equation $\frac{dy}{dx}=\frac{x^2+3y^2}{3yx}$ is homogeneous, since it is written in the standard form $\frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and both are homogeneous functions of the same degree

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Use the substitution: $y=ux$

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Expand and simplify

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

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Solve the integral $\int3udu$ and replace the result in the differential equation

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

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Replace $u$ with the value $\frac{y}{x}$

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The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

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Multiplying fractions $\frac{3}{2} \times \frac{y^2}{x^2}$

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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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