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

Multiplying the fraction by $-1$

Multiply both sides of the equation by $dx$

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

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

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

Use the substitution: $y=ux$

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

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

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