To solve the equation \((y')^2 + xy = y^2 + x(y')\), let's rewrite it with \( y' = \frac{dy}{dx} \) for clarity:

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

This equation is a nonlinear first-order differential equation. Let’s proceed by simplifying and examining if there is a way to solve it directly or by substituting terms.

### Step-by-Step Solution

1. **Rewrite the Equation in Terms of \( y' \):**
   Expand and rearrange the terms as:
   \[
   \left( \frac{dy}{dx} \right)^2 - x \frac{dy}{dx} + xy - y^2 = 0
   \]

2. **Consider it as a Quadratic in \( \frac{dy}{dx} \):**
   Rewrite the equation as:
   \[
   \left( \frac{dy}{dx} \right)^2 - x \frac{dy}{dx} + (xy - y^2) = 0
   \]
   This is now a quadratic equation in \( \frac{dy}{dx} \), which we can denote by \( u = \frac{dy}{dx} \).

3. **Solve for \( u \):**
   Treat this as a quadratic equation \( u^2 - xu + (xy - y^2) = 0 \) in \( u \):
   \[
   u = \frac{dy}{dx} = \frac{x \pm \sqrt{x^2 - 4(xy - y^2)}}{2}
   \]
   Simplifying further would involve analyzing the expression under the square root, \( x^2 - 4(xy - y^2) \), which may not lead to elementary functions.

In this form, the equation appears complex and likely does not yield a straightforward analytical solution. However, we could approach it numerically or consider a substitution method if more details were available on the form of \( y \) or \( x \).

Would you like further details on substitution techniques, or any specific numerical approach?

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### Related Questions

1. How can we use substitution methods to simplify differential equations?
2. What are the common techniques for solving nonlinear differential equations?
3. How does one interpret a quadratic in terms of derivatives?
4. Are there cases where differential equations have no closed-form solutions?
5. What numerical methods are suitable for solving complex differential equations?

### Tip:
In differential equations, recognizing patterns can help reduce complexity, especially for nonlinear forms.

Explore a step-by-step solution to the nonlinear differential equation (y')^2 + xy = y^2 + x(y'). Learn to simplify complex equations by rewriting and solving as a quadratic in derivatives, using substitution methods, and analyzing solution approaches.

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