Question

The differential equation for which \( y^2 = 4a(x + a) \) (where \( a \) is a parameter) is the general solution is:

• A. \( y = 2x \frac{dy}{dx} + y \left(\frac{dy}{dx}\right)^2 \)
• B. \( y = y \frac{dy}{dx} - x \left(\frac{dy}{dx}\right)^2 \)
• C. \( x = 3 \frac{dy}{dx} + y \left(\frac{dy}{dx}\right)^2 \)
• D. \( y = 3x \frac{dy}{dx} + y^2 \left(\frac{dy}{dx}\right)^2 \)

Hint:

For parameter-based differential equations, differentiate given expressions explicitly, solve for \( \frac{dy}{dx} \), and rewrite in required form.

Answer 1

The Correct Option is A

Given: \[ y^2 = 4a(x + a) \] Differentiating both sides with respect to \( x \): \[ 2y \frac{dy}{dx} = 4a \] Solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \] Expressing in terms of the differential equation: \[ y = 2x \frac{dy}{dx} + y \left(\frac{dy}{dx}\right)^2 \] Thus, the correct answer is: \[ y = 2x \frac{dy}{dx} + y \left(\frac{dy}{dx}\right)^2 \]