Find the general solution of the differential equation

y+y3=(y5+2x)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 given the differential equation:

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

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

We start by multiplying both sides by \(dx\) to convert the equation into differential form:

\[
(y + y^3) dx = (y^5 + 2x) dy.
\]

This form is now separable. Next, we proceed to separate the variables.

### Step 2: Separate the variables

We need to get all terms involving \(x\) on one side of the equation and all terms involving \(y\) on the other side. Rearranging the equation, we get:

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

This step, however, does not simplify easily. Instead, we check for potential simplifications. Notice that the equation doesn't immediately allow separation of variables, and exact integration isn't straightforward. So, we seek to rewrite the equation in an implicit form where a general solution can be found.

Let me check that first approach.We have the differential equation in the form:

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

Since direct separation is difficult here, the equation hints at potential integration through substitution or a more implicit method. We aim to rewrite this in a form where integration on both sides becomes possible or to express it implicitly.

Let me now work toward expressing the solution \(F(x, y) = C\).The differential equation, \( y + y^3 = (y^5 + 2x) \frac{dy}{dx} \), cannot be easily separated, so let's attempt to solve it implicitly by rewriting it into an exact differential form.

We'll proceed by checking if the differential form:

\[
(y + y^3) dx = (y^5 + 2x) dy
\]

is an exact equation. If it's exact, there will exist a function \( F(x, y) \) such that:

\[
dF = M(x, y) dx + N(x, y) dy
\]

where:
- \( M(x, y) = y + y^3 \)
- \( N(x, y) = y^5 + 2x \)

Next, let's check if this equation is exact by verifying whether \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). 

I'll calculate these partial derivatives and see if the equation is exact.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you want, I can explain how to continue solving it manually or provide further guidance on what we've done so far. Would you like more details or assistance?

Here are 5 questions to help expand on the current work:
1. How do you determine if a differential equation is exact?
2. What does it mean for a differential equation to be separable?
3. How can you apply substitution methods in non-exact equations?
4. What are the methods to solve implicit equations in calculus?
5. How can you check whether a differential form is integrable?

**Tip**: When solving differential equations, check for exactness by verifying if the mixed partial derivatives of the functions involved are equal.

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

Learn how to find the general solution of the differential equation y + y^3 = (y^5 + 2x)y' using methods such as exact equations and separation of variables. This problem involves key concepts in differential equations and is suitable for undergraduate students.

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