Step-by-step Solution

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

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Final Answer

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

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

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 $\frac{3yx^3}{x}$ 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 $C$

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

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

$y=\frac{\frac{1}{2}x^2+C_0}{x^3}$
11

Find the explicit solution to the differential equation

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

Final Answer

$y=\frac{\frac{1}{2}x^2+C_0}{x^3}$
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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Tips on how to improve your answer:

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

Related Formulas:

1. See formulas

Time to solve it:

~ 0.11 s