Find the general solution of the differential equation
y
+
1
y
3
=
(
y
5
+
2
x
)
y
'
Write your solution in the form 
F
(
x
,
y
)
=
C
,
 ﻿where 
C
 ﻿is an arbitrary constant.
(
1
)
 
=
C
.
Hint: Start by rewriting the equation in differential form.

We are tasked with finding the general solution to the differential equation:

\[
y + \frac{1}{y^3} = (y^5 + 2x)y'
\]

and expressing the solution in the form:

\[
F(x, y) = C
\]

where \(C\) is an arbitrary constant.

### Step 1: Rewrite the equation in differential form

First, let's separate the terms involving \(y\) on one side and \(x\) on the other. To do this, we start by expressing the equation in differential form. Using \( y' = \frac{dy}{dx} \), the given equation can be rewritten as:

\[
y + \frac{1}{y^3} = (y^5 + 2x) \frac{dy}{dx}
\]

Multiply both sides by \(dx\):

\[
\left(y + \frac{1}{y^3}\

Find the general solution of the differential equation y + 1/y^3 = (y^5 + 2x)y'. Write your solution in the form F(x, y) = C, where C is an arbitrary constant. Hint: Start by rewriting the equation in differential form.

Find the general solution to the differential equation y + 1/y^3 = (y^5 + 2x)y' using the method of separation of variables. This problem involves key concepts from differential equations and is suitable for university-level calculus or differential equations courses.

Math Problem Statement

Find the general solution of the differential equation y + 1 y 3

( y 5 + 2 x ) y ' Write your solution in the form F ( x , y )

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