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.

Learn how to solve the transcendental equation x^2y = xy + x^2e^y/x using algebraic manipulation and factoring techniques. Understand the concept of transcendental equations and explore methods for solving them numerically or graphically. Suitable for students in Grades 10-12.

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