To solve the equation $x^2y = xy + \frac{x^2e^y}{x}$, let's first simplify the expression.

The equation is:

$x^2y = xy + \frac{x^2e^y}{x}$

We can simplify the right-hand side:

$x^2y = xy + x e^y$

Now, let's rearrange the equation to bring all terms involving $y$ to one side:

$x^2y - xy = x e^y$

Factor out $y$ from the left-hand side:

$y(x^2 - x) = x e^y$

Now, solve for $y$:

$y = \frac{x e^y}{x^2 - x}$

This equation involves both $y$ and $e^y$, making it transcendental. There isn't a straightforward algebraic solution for such equations, but we can explore solving it numerically or graphically.

Would you like to proceed with a numerical solution, graph the functions, or explore further?

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5 Questions to Expand Understanding:

1. What is a transcendental equation, and how does it differ from algebraic equations?
2. Can we apply the Lambert W function to solve equations of the form $y e^y = x$?
3. How do numerical methods like Newton-Raphson work for solving equations?
4. What is the significance of factoring in algebraic manipulation?
5. How can graphing functions help us understand the nature of their solutions?

Tip: When dealing with transcendental equations, understanding graphical behavior or using numerical methods is often key to finding solutions.