** 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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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 $x^3$

Any expression multiplied by $1$ is equal to itself

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

Multiplying the fraction by $x^2$

Combine all terms into a single fraction with $2$ as common denominator

We can rename $2\cdot C_0$ as other constant

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

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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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