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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