# Step-by-step Solution

## Solve the differential equation $\frac{dy}{dx}+y\frac{3}{x}=\frac{1}{x^2}$

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### Videos

$y=x^{-3}\left(\frac{1}{2}x^2+C_0\right)$

## Step-by-step Solution

Problem to solve:

$\frac{dy}{dx}+\frac{3}{x}y=\frac{1}{x^2}$
1

Multiplying the fraction by $y$

$\frac{dy}{dx}+\frac{3y}{x}=\frac{1}{x^2}$
2

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 $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$

Compute the integral

$\int\frac{3}{x}dx$

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

$3\ln\left(x\right)$
3

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

$\int P(x)dx=\int\frac{3}{x}dx=3\ln\left(x\right)$

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

$x^3$
4

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

$\mu(x)=x^3$

Multiplying the fraction by $x^3$

$x^3\frac{dy}{dx}+\frac{3yx^3}{x}=\frac{x^3}{x^2}$

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

$x^3\frac{dy}{dx}+\frac{3yx^3}{x}=x$

Simplify the fraction by $x$

$x^3\frac{dy}{dx}+3yx^{2}=x$
5

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

$x^3\frac{dy}{dx}+3yx^{2}=x$
6

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

$\frac{d}{dx}\left(x^3y\right)=x$
7

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

$\int\frac{d}{dx}\left(x^3y\right)dx=\int xdx$
8

Simplify the left side of the differential equation

$x^3y=\int xdx$

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 $n=1$

$\frac{1}{2}x^2$
9

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

$x^3y=\frac{1}{2}x^2$
10

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

$x^3y=\frac{1}{2}x^2+C_0$

Multiply the equation by the reciprocal of $x^3$

$y=x^{-3}\left(\frac{1}{2}x^2+C_0\right)$
11

Find the explicit solution to the differential equation

$y=x^{-3}\left(\frac{1}{2}x^2+C_0\right)$

$y=x^{-3}\left(\frac{1}{2}x^2+C_0\right)$
$\frac{dy}{dx}+\frac{3}{x}y=\frac{1}{x^2}$

### Main topic:

Differential Equations

~ 0.08 s