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

Step 1: Note the number of constants.
The family $y=ax^2-2abx+ab^2$ carries two arbitrary constants $a$ and $b$, so we expect a second order differential equation after eliminating them.
Step 2: Recognise a perfect square.
Rewrite the family as $y=a(x^2-2bx+b^2)=a(x-b)^2$. This compact form makes elimination easy.
Step 3: Differentiate once.
$\dfrac{dy}{dx}=2a(x-b)$.
Step 4: Differentiate again.
$\dfrac{d^2y}{dx^2}=2a$.
Step 5: Eliminate the constants.
From the first derivative, $\left(\dfrac{dy}{dx}\right)^2=4a^2(x-b)^2=4a\cdot a(x-b)^2=4a\,y$ since $y=a(x-b)^2$.
Step 6: Replace $a$.
Because $\dfrac{d^2y}{dx^2}=2a$, we have $4a=2\dfrac{d^2y}{dx^2}$, so $\left(\dfrac{dy}{dx}\right)^2=2y\dfrac{d^2y}{dx^2}$.
\[ \boxed{2y\dfrac{d^2y}{dx^2}=\left(\dfrac{dy}{dx}\right)^2} \]