Find a first-order differential equation having the family
![\[ xy = C \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-267e7edd62b0b3a1715189cdbee82223_l3.png "Rendered by QuickLaTeX.com")
as integral curves.
Since 
we can differentiate both sides with respect to 
to get
![\[ xy = C \quad \implies \quad xy' + y = 0. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c7ace290a377e30f4d4b52ca8a9fa9e8_l3.png "Rendered by QuickLaTeX.com")
Thus, the first-order differential equation
![\[ xy' + y = 0 \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0f084db6d87c79e413191d4d3e33eefc_l3.png "Rendered by QuickLaTeX.com")
has the given family as integral curves.
Find a first-order differential equation having the family
as integral curves.
Since we can differentiate both sides with respect to to get
Thus, the first-order differential equation
has the given family as integral curves.