Question

If \(a,b\) are arbitrary constants, then the differential equation corresponding to family \[ y=ax^2-2abx+ab^2 \] is

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

Hint:

Number of arbitrary constants determines order of required differential equation.

Answer 1

The Correct Option is B

Concept: To form differential equation: 1. Differentiate enough times to eliminate arbitrary constants 2. Express final relation only in terms of \(x,y\) and derivatives. Two arbitrary constants imply second order differential equation.

Step 1: Differentiate once.
Given \[ y=ax^2-2abx+ab^2 \] Differentiate. \[ y' = 2ax-2ab \]

Step 2: Differentiate second time.
\[ y''=2a \] Thus \[ a=\frac{y''}{2} \] From first derivative \[ y' = 2a(x-b) \] \[ = y''(x-b) \] Hence \[ b=x-\frac{y'}{y''} \]

Step 3: Substitute into original equation.
Substituting carefully and eliminating constants gives \[ 2yy''=(y')^2 \] Thus \[ \boxed{ 2y\frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2 } \]