To solve the differential equation

\[
\frac{dy}{dx} + y = xy^4
\]

we’ll first rewrite it in a form that allows us to use separation of variables or an integrating factor if possible.

1. Rewrite the equation:
   \[
   \frac{dy}{dx} = xy^4 - y
   \]

2. Factor out \( y \) on the right-hand side:
   \[
   \frac{dy}{dx} = y(xy^3 - 1)
   \]

Now, we can attempt to use separation of variables, but because of the \(y^4\) term, let’s check if this is a Bernoulli equation.

### Step 1: Recognize the Bernoulli Form
The general form for a Bernoulli equation is:
\[
\frac{dy}{dx} + P(x)y = Q(x)y^n
\]
In our case:
- \( P(x) = 1 \)
- \( Q(x) = x \)
- \( n = 4 \)

Since \( n \neq 1 \), this is indeed a Bernoulli equation.

### Step 2: Substitute to Simplify
To solve, let’s use the substitution:
\[
v = y^{1-n} = y^{1-4} = y^{-3}
\]
Then,
\[
\frac{dv}{dx} = -3y^{-4} \frac{dy}{dx}
\]
or
\[
\frac{dy}{dx} = -\frac{1}{3}y^4 \frac{dv}{dx}
\]

Substituting these expressions into the original differential equation, we can proceed with solving for \( v(x) \) and then return to find \( y(x) \). 

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**Related Questions to Explore:**

1. How do you solve a general Bernoulli differential equation?
2. What are other methods to approach nonlinear differential equations?
3. Can an integrating factor be used for nonlinear equations?
4. How do you verify if a differential equation is exact?
5. What other types of substitutions are useful in solving differential equations?

**Tip:** For Bernoulli equations, the substitution \( v = y^{1-n} \) transforms it into a linear differential equation, making it easier to solve.

Explore the solution of the differential equation dy/dx + y = xy^4. This guide uses the Bernoulli equation method, covering key concepts in nonlinear differential equations, substitutions, and solution transformations. Ideal for advanced high school and college students.

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