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

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##  Final answer to the problem

$y=\frac{x^2+C_1}{2x^3}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Exact Differential Equation
• Linear 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
Can't find a method? Tell us so we can add it.
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 $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$
3

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

$\int P(x)dx=\int\frac{3}{x}dx=3\ln\left|x\right|$
4

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

$\mu(x)=x^3$
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5

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

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

$x^3y=\int xdx$
9

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

$x^3y=\frac{1}{2}x^2+C_0$
10

Find the explicit solution to the differential equation. We need to isolate the variable $y$

$y=\frac{x^2+C_1}{2x^3}$

##  Final answer to the problem

$y=\frac{x^2+C_1}{2x^3}$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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4
5
6
7
8
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

###  Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.