To solve the equation \( ye\left(\frac{y^2}{x}\right) = Cx \) and manipulate it to get the desired result, we can take the following steps:

1. **Step 1: Start with the original equation.**

   \[
   ye\left(\frac{y^2}{x}\right) = Cx
   \]

2. **Step 2: Isolate the exponential term.**

   Divide both sides of the equation by \( y \) (assuming \( y \neq 0 \)):

   \[
   e\left(\frac{y^2}{x}\right) = \frac{Cx}{y}
   \]

3. **Step 3: Take the natural logarithm on both sides.**

   Apply \( \ln \) to both sides to remove the exponential:

   \[
   \frac{y^2}{x} = \ln\left(\frac{Cx}{y}\right)
   \]

4. **Step 4: Multiply both sides by \( x \).**

   Multiply both sides of the equation by \( x \) to eliminate the fraction:

   \[
   y^2 = x \ln\left(\frac{Cx}{y}\right)
   \]

This is a simplified version of the equation, though it might not directly resemble your target form \( ye(y^2/x) = Cx \). If you wanted further manipulation or explanation on a specific form, feel free to ask!

Would you like more details, or do you have specific questions about this solution?

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Here are 5 related questions to expand your understanding:

1. How do natural logarithms work when manipulating exponential equations?
2. What happens if \( y = 0 \) in the original equation?
3. Could the equation be solved explicitly for \( y \) in terms of \( x \)?
4. How does the constant \( C \) affect the general solution?
5. What would happen if we used a different base for the exponential function?

**Tip**: Always check domain restrictions, especially when dealing with logarithms and exponentials!

Explore how to solve and manipulate the exponential equation ye(y^2/x) = Cx step-by-step, using properties of exponents and logarithms. This solution highlights key mathematical concepts such as natural logarithms and exponential functions, suitable for college-level students.

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