If the solution of the differential equation $x (\frac{d^{3} y}{d x^{3}}) + \frac{d^{2} y}{d x^{2}} + 2 x (\frac{d y}{d x}) - y + 1 = 0$ is a straight line of the form $y = m x + c,$ where $m, c$ are real constants, then the number of such possible lines is

1

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2

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3

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infinite

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Solution

The correct option is A $1$
Given that the equation of line $y = m x + c$
$y^{'} = m, y^{''} = 0, y^{'''} = 0$
$\Rightarrow x \times 0 + 0 + 2 \cdot x \cdot m - m x - c + 1 = 0 \Rightarrow m x - c + 1 = 0 \Rightarrow m = 0, c = 1$

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