Q.Find the differential equation of the family of all straight lines passing through the origin.

$\dfrac{dy}{dx} = \dfrac{y}{x}$.

Step 1 — equation of the family. Every straight line through the origin has the form

$y = mx,$

where $m$ is the arbitrary constant (the slope).

Step 2 — differentiate to remove the constant.

$\frac{dy}{dx} = m.$

Step 3 — eliminate $m$. From $y = mx$ we have $m = \dfrac{y}{x}$, so

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