We are given the differential equation:

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

To solve this, let's rewrite it in a standard form:

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

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

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

This is a Bernoulli equation of the form:

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

In our case, \(n = 2\), \(P(x) = \frac{1 + x}{x}\), and \(Q(x) = 1\).

To solve a Bernoulli equation, we first divide through by \(y^2\) to make it linear. We substitute \(v = y^{-1}\), which gives \( \frac{dv}{dx} = -y^{-2} \frac{dy}{dx} \).

After this transformation, the equation becomes linear, which we can solve. Would you like me to continue with the full solution step-by-step?

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Here are 5 questions for further exploration:
1. What are Bernoulli equations?
2. How do we transform non-linear differential equations to linear ones?
3. Can we solve this differential equation using an integrating factor?
4. What is the general solution to Bernoulli's equation?
5. How do different values of \(n\) change the form of the solution for Bernoulli's equation?

**Tip:** Always check if a differential equation can be reduced to a simpler form before attempting to solve directly!

Explore the step-by-step solution to the differential equation x(dy/dx) - (1+x)y = xy^2 using Bernoulli's method. Learn how to transform non-linear differential equations into linear forms, suitable for undergraduate calculus students.

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