Form the differential equation whose general solution is 𝑦 = 4 ( 𝑥 −𝐴)^ 2, where A is arbitrary constants

To form the differential equation from the general solution \( y = 4(x - A)^2 \), we need to eliminate the arbitrary constant \( A \).

### Steps:

1. **Express \( A \) in terms of \( x \) and \( y \):**

   \[
   y = 4(x - A)^2
   \]
   Divide both sides by 4:
   \[
   \frac{y}{4} = (x - A)^2
   \]
   Take the square root of both sides:
   \[
   \sqrt{\frac{y}{4}} = x - A
   \]
   Simplify:
   \[
   \frac{\sqrt{y}}{2} = x - A
   \]
   So, \( A \) can be written as:
   \[
   A = x - \frac{\sqrt{y}}{2}
   \]

2. **Differentiate \( y \) with respect to \( x \):**

   Differentiate both sides of the equation \( y = 4(x - A)^2 \) with respect to \( x \):

   \[
   \frac{dy}{dx} = 4 \cdot 2(x - A) \cdot \frac{d}{dx}(x - A)
   \]
   Simplifying:
   \[
   \frac{dy}{dx} = 8(x - A) \cdot \left(1 - \frac{dA}{dx}\right)
   \]
   
   But since \( A = x - \frac{\sqrt{y}}{2} \), then:
   \[
   \frac{dA}{dx} = 1 - \frac{1}{4\sqrt{y}} \cdot \frac{dy}{dx}
   \]

   Substituting \( \frac{dA}{dx} \) back into the derivative:
   \[
   \frac{dy}{dx} = 8(x - A) \cdot \left(1 - \left(1 - \frac{1}{4\sqrt{y}} \cdot \frac{dy}{dx}\right)\right)
   \]
   Simplifying:
   \[
   \frac{dy}{dx} = \frac{8(x - A)}{4\sqrt{y}} \cdot \frac{dy}{dx}
   \]
   Therefore:
   \[
   \frac{dy}{dx} = \frac{2y}{\sqrt{y}} \cdot \frac{dy}{dx}
   \]
   Cancelling \( \frac{dy}{dx} \) from both sides:
   \[
   1 = \frac{2\sqrt{y}}{4\sqrt{y}}
   \]
   Finally, we obtain the differential equation:
   \[
   \frac{dy}{dx} = 4\sqrt{y}
   \]

### Final Answer:
The differential equation whose general solution is \( y = 4(x - A)^2 \) is:
\[
\frac{dy}{dx} = 4\sqrt{y}
\]

Would you like more details on any step? Here are 5 questions to deepen your understanding:

1. How do we differentiate the general solution with respect to \( x \)?
2. What does the term "eliminate the arbitrary constant" mean in this context?
3. What role does the square root play in finding the differential equation?
4. How do we simplify \( A \) in terms of \( x \) and \( y \)?
5. How does this method apply to other forms of general solutions?

**Tip:** Understanding how to differentiate implicitly can help greatly when eliminating arbitrary constants.

Learn how to derive a differential equation from the general solution y = 4(x - A)^2 step-by-step. This problem involves differential equations and implicit differentiation, essential for advanced undergraduate students and beyond.

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