## Final Answer

## Step-by-step explanation

Problem to solve:

Multiplying the fraction by $y$

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 $P(x)=\frac{3}{x}$ and $Q(x)=\frac{1}{x^2}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

Compute the integral

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

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

Simplify $e^{3\ln\left(x\right)}$ by applying the properties of exponents and logarithms

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

Multiplying the fraction by $x^3$

Simplify the fraction $\frac{x^3}{x^2}$ by $x$

Simplify the fraction by $x$

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

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

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

Simplify the left side of the differential equation

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 $n=1$

Solve the integral $\int xdx$ and replace the result in the differential equation

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

Divide both sides of the equation by $x^3$