For the following, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

$\frac{x}{a} + \frac{y}{b} = 1$

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Solution

Given, family is $\frac{x}{a} + \frac{y}{b} = 1 . . . (i)$
On differentiating both sides w.r.t. x, we get $\frac{1}{a} + \frac{1}{b} \frac{d y}{d x} = 0$
Again differentiating both sides w.r.t. x, we get
$0 + \frac{1}{b} \frac{d^{2} y}{d x^{2}} = 0 = y^{''} = 0.$
which is the required differential equation.
Note: Differentiate the given equation as many number of times as the number of arbitrary constants present in the equation.

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