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.