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.