The given differential equation is:

$\frac{dy}{dx} = y(xy^4 - 1)$

Step 1: Identify Bernoulli’s Equation Form

A Bernoulli equation has the standard form:

$\frac{dy}{dx} + P(x)y = Q(x)y^n$

Rewriting the given equation:

$\frac{dy}{dx} = xy^5 - y$

$\frac{dy}{dx} + y = xy^5$

Comparing with the standard form, we identify:

• $P(x) = 1$
• $Q(x) = x$
• $n = 5$ (since $y^5$ appears on the right-hand side)

Step 2: Use the Bernoulli Substitution

For a Bernoulli equation, we use the substitution:

$v = y^{1-n} = y^{1-5} = y^{-4}$

Differentiating both sides with respect to $x$:

$\frac{dv}{dx} = -4y^{-5} \frac{dy}{dx}$

Substituting $\frac{dy}{dx}$ from the given DE:

$\frac{dv}{dx} = -4y^{-5} (xy^5 - y)$

$\frac{dv}{dx} = -4(xy^0 - y^{-4})$

$\frac{dv}{dx} = -4(x - v)$

Rearranging:

$\frac{dv}{dx} + 4v = 4x$

Step 3: Solve the Linear Differential Equation

This is a first-order linear DE of the form:

$\frac{dv}{dx} + P(x)v = Q(x)$

where $P(x) = 4$ and $Q(x) = 4x$. The integrating factor (IF) is:

$IF = e^{\int 4dx} = e^{4x}$

Multiplying both sides by $e^{4x}$:

$e^{4x} \frac{dv}{dx} + 4e^{4x} v = 4x e^{4x}$

Recognizing the left-hand side as the derivative of a product:

$\frac{d}{dx} (v e^{4x}) = 4x e^{4x}$

Integrating both sides:

$v e^{4x} = \int 4x e^{4x} dx$

Using integration by parts with $u = x$ and $dv = 4e^{4x}dx$:

$\int 4x e^{4x} dx = x e^{4x} - \frac{1}{4} e^{4x} + C$

$= e^{4x} \left(x - \frac{1}{4}\right) + C$

Thus:

$v e^{4x} = e^{4x} \left(x - \frac{1}{4}\right) + C$

$v = x - \frac{1}{4} + C e^{-4x}$

Step 4: Substitute Back $v = y^{-4}$

$y^{-4} = x - \frac{1}{4} + C e^{-4x}$

$y = \left(x - \frac{1}{4} + C e^{-4x} \right)^{-\frac{1}{4}}$

Final Answer:

$y(x) = \left(x - \frac{1}{4} + C e^{-4x} \right)^{-\frac{1}{4}}$

Would you like a step-by-step breakdown of integration by parts or any other clarification?

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Related Questions:

1. What is a Bernoulli equation, and how do you recognize it?
2. How do you determine the appropriate substitution for Bernoulli’s equation?
3. Can you solve a Bernoulli equation without using an integrating factor?
4. How does the integrating factor method work for linear differential equations?
5. What are other examples of nonlinear differential equations that can be transformed into linear ones?

Tip: Always check if a given differential equation can be transformed into a simpler form before attempting to solve it directly!