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Find a first-order differential equation having the family 2x + 3y = C as integral curves

Find a first-order differential equation having the family

    \[ 2x + 3y = C \]

as integral curves.


Since 2x+3y = C we can differentiate both sides with respect to x to get

    \[ 2x+3y = C \quad \implies \quad 3y' + 2 = 0 \quad \implies \quad y' + \frac{2}{3} = 0. \]

Thus, the first-order differential equation

    \[ y' + \frac{2}{3} = 0 \]

has the given family as integral curves.

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