Question

The differential equation obtained from the function \( y = a(x - a)^2 \) is

• A. \( 8y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x - \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \)
• B. \( 8y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x + \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \)
• C. \( 2y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x - \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \)
• D. \( 4y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x - \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \)

Hint:

When differentiating functions of the form \( y = a(x - a)^2 \), use the chain rule and substitute the expressions back into the equation to obtain the desired form.

Answer 1

The Correct Option is D

Step 1: Differentiating the function.
The given function is \( y = a(x - a)^2 \). We differentiate this with respect to \( x \): \[ \frac{dy}{dx} = 2a(x - a) \]
Step 2: Substituting into the equation.
We substitute \( \frac{dy}{dx} = 2a(x - a) \) into the equation \( y = a(x - a)^2 \) to get the differential equation. After simplifying, we obtain the equation \( 4y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x - \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \).

Step 3: Conclusion.
Thus, the correct differential equation is \( 4y^2 = \left( \frac{dy}{dx} \right)^2 \left[ x - \frac{1}{4y} \left( \frac{dy}{dx} \right)^2 \right] \), which makes option (D) the correct answer.