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Form the differential equation by eliminating the arbitrary constant from the equation $y = a x^{3} + b x^{2}$

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$\begin{aligned} x^{2} \frac{d^{2} y}{d x^{2}} - 4 x \frac{d y}{d x} - 6 y = 0 \end{aligned}$

$\begin{aligned} \frac{d^{2} y}{d x^{2}} - 4 x \frac{d y}{d x} + 6 y = 0 \end{aligned}$

$x^{2} \frac{d^{2} y}{d x^{2}} + 4 x \frac{d y}{d x} + 6 y = 0$

$x^{2} \frac{d^{2} y}{d x^{2}} - 4 x \frac{d y}{d x} + 6 y = 0$

answer is D.

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Detailed Solution

Given $y = a x^{3} + b x^{2}, y_{1} = 3 a x^{2} + 2 b x, y_{2} = 6 a x + 2 b \Rightarrow x y_{2} - y_{1} = 3 a x^{2}, 2 y_{1} - x y_{2} = 2 b x$

$6 y = 6 a x^{3} + 6 b x^{2} = 2 x [x y_{2} - y_{1}] + 3 x [2 y_{1} - x y_{2}] = 2 x^{2} y_{2} - 2 x y_{1} + 6 x y_{1} - 3 x^{2} y_{2}$

$\Rightarrow x^{2} y_{2} - 4 x y_{1} + 6 y = 0$

$∴$ The required differential equation $x^{2} \frac{d^{2} y}{d x^{2}} - 4 x \frac{d y}{d x} + 6 y = 0$

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