Question

By eliminating the arbitrary constants $A$ and $B$ from $y = Ax^{2} + Bx$, the differential equation obtained is

• A. $\dfrac{d^{2}y}{dx^{2}} = 0$
• B. $x^{2}\dfrac{d^{2}y}{dx^{2}} - 2x\dfrac{dy}{dx} + 2y = 0$
• C. $\dfrac{d^{2}y}{dx^{2}} = 0$
• D. $x^{2}\dfrac{d^{2}y}{dx^{2}} + y = 0$

Hint:

To form a differential equation from a family with $n$ arbitrary constants, differentiate $n$ times then eliminate the constants algebraically.

Answer 1

The Correct Option is B

Step 1: Conceptual Understanding:
Differentiate twice to generate two extra equations and eliminate both constants.
Step 2: Explanation in Detail:
$y = Ax^2 + Bx$, $\;y' = 2Ax + B$, $\;y'' = 2A$.
From $y''$: $A = \dfrac{y''}{2}$. From $y'$: $B = y' - 2Ax = y' - xy''$.
Substituting into $y$: $y = \dfrac{y''}{2}x^2 + (y'-xy'')x = xy' - \dfrac{x^2y''}{2}$.
Multiply by $-2$: $x^2y'' - 2xy' + 2y = 0$.
Step 3: Therefore, Stating the Final Answer
$x^2\dfrac{d^2y}{dx^2} - 2x\dfrac{dy}{dx} + 2y = 0$.