Question

Form a differential equation representing the given family of curves by eliminating arbitrary constants $a$ and $b$
$\frac{x}{a}+\frac{y}{b}=1$.

Answer 1

Let the curve

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

Differentiating both sides w.r.t. x
we get,

$\frac{1}{a}+\frac{1}{b}.\frac{dy}{dx}=0$

$\frac{1}{b}.\frac{dy}{dx}=\frac{-1}{a}$

$\frac{dy}{dx}=\frac{-b}{a}$

Again, differentiating both sides w.r.t. x
we get,

$\frac{d^{2}y}{d{x}^2}=\frac{d\left(\frac{-b}{a}\right)}{dx}$

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

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

Final Answer:
Hence, the required differential equation is

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