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# Find a first-order differential equation having the family y2 = Cx 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

Then, since we have . Substituting this on the right hand side we have the first-order differential equation

having the given family as integral curves.

# Find a first-order differential equation having the family xy = C 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.

# Find a first-order differential equation having the family x2 – y2 = c 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.

# Find a first-order differential equation having the family y = Ce-2x 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

Then, we plug in our original equation on the right hand side to get the first-order differential equation

having the given family as integral curves.

# Find a first-order differential equation having the family 2x + 3y = C 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.