** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

We could not solve this problem by using the method: **Exact Differential Equation**

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Multiplying the fraction by $y$

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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 $P(x)=\frac{3}{x}$ 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, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

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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

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

Multiplying the fraction by $x^3$

Multiplying the fraction by $x^3$

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

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

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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

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 $n=1$

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

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

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

Simplifying the quotients

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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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