Find one-parameter families of solution curves of the following differential equations:

Given

This is a first order linear differential equation of the form

Here, P = –1 and Q = y

The integrating factor (I.F) of this differential equation is,

We have

∴ I.F = e^–y

Hence, the solution of the differential equation is,

Recall

⇒ xe^–y = –ye^–y – e^–y + c

⇒ xe^–y = –e^–y(y + 1) + c

⇒ xe^–y × e^y = [–e^–y(y + 1) + c] × e^y

∴ x = –(y + 1) + ce^y

Thus, the solution of the given differential equation is x = –(y + 1) + ce^y