Obtain the differential equation of which $y = a x^{2} + b x$ is the general solution ,a,b being arbitrary constants.

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

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x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} + 2 y = 0

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x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} - 2 y = 0

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

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Solution

The correct option is A $x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} + 2 y = 0$
$y = a x^{2} + b x \dots (1)$
$y_{1} = 2 a x + b \dots (2)$
$y_{2} = 2 a \dots (3)$
Eliminating $b$ between (1)and (2),we get
$x y_{1} - y = a x^{2} a x^{2} = \frac{1}{2} x^{2} y_{2}, b y (3)$
$∴ x^{2} \frac{d^{2} y}{d x^{2}} - 2 x \frac{d y}{d x} + 2 y = 0$

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