To solve the differential equation:

\[
y' = \frac{x+1}{y^4+1}
\]

We can approach this as a separable differential equation.

### Step 1: Rewrite in separable form
Separate the variables \(y\) and \(x\):

\[
(y^4 + 1) \, dy = (x+1) \, dx
\]

### Step 2: Integrate both sides
Now integrate both sides:

\[
\int (y^4 + 1) \, dy = \int (x+1) \, dx
\]

#### Left-hand side:

\[
\int (y^4 + 1) \, dy = \frac{y^5}{5} + y
\]

#### Right-hand side:

\[
\int (x+1) \, dx = \frac{x^2}{2} + x
\]

### Step 3: Combine the results
Now, combine the results from both integrals:

\[
\frac{y^5}{5} + y = \frac{x^2}{2} + x + C
\]

where \(C\) is the constant of integration.

### Step 4: Final equation
This is the implicit solution to the differential equation:

\[
\frac{y^5}{5} + y = \frac{x^2}{2} + x + C
\]

Let me know if you need further details or clarification!

Here are some follow-up questions you might find helpful:
1. How can we solve for \( y \) explicitly?
2. How does the constant \(C\) affect the solution?
3. What are the boundary or initial conditions for this problem?
4. Can we classify this equation as a particular type of ODE?
5. How would numerical methods solve this?

**Tip:** For separable differential equations, always ensure variables are properly separated before integrating.

This guide provides a step-by-step solution to the separable differential equation y' = (x + 1) / (y^4 + 1). It covers key mathematical concepts like integration and separation of variables, suitable for students in undergraduate calculus or differential equations courses.

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