Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

If $y=a+b x^{2}, a, b$ arbitrary constants, then

A. $\frac{d^{2} y}{d x^{2}}=2 x y$

B. $x \frac{d^{2} y}{d x^{2}}=y_{1}$

C. $x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+y=0$

D. $x \frac{d^{2} y}{d x^{2}}=2 x y$

Solution:

Given:

$y=a+b x^{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{bx}$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=2 \mathrm{~b} \neq 2 \mathrm{xy}$

$x \frac{d^{2} y}{d x^{2}}=2 b x$

$=\frac{\mathrm{dy}}{\mathrm{dx}}$

$x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+y=2 b x-2 b x+y$

$=\mathrm{y}$

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