Question

The differential equation satisfied by the system of parabolas y² = 4a(x + a) is :

• A. y (dy/dx)² + 2x (dy/dx) - y = 0
• B. y (dy/dx)² - 2x (dy/dx) + y = 0
• C. y (dy/dx)² - 2x (dy/dx) - y = 0
• D. y (dy/dx)² - 2x (dy/dx) = 0

Hint:

To form a differential equation, differentiate the family of curves and eliminate the arbitrary constant ($a$).

Answer 1

The Correct Option is A

Step 1: $y^2 = 4ax + 4a^2$. Differentiate with respect to $x$: $2y \frac{dy}{dx} = 4a \implies a = \frac{y}{2} \frac{dy}{dx}$.
Step 2: Substitute $a$ back into the original equation: $y^2 = 4 \left( \frac{y}{2} \frac{dy}{dx} \right) x + 4 \left( \frac{y}{2} \frac{dy}{dx} \right)^2$.
Step 3: $y^2 = 2xy \frac{dy}{dx} + y^2 \left( \frac{dy}{dx} \right)^2$.
Step 4: Divide by $y$ (assuming $y \neq 0$): $y = 2x \frac{dy}{dx} + y \left( \frac{dy}{dx} \right)^2$.
Step 5: Rearrange: $y \left( \frac{dy}{dx} \right)^2 + 2x \left( \frac{dy}{dx} \right) - y = 0$.