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 × ey = [–e–y(y + 1) + c] × ey
∴ x = –(y + 1) + cey
Thus, the solution of the given differential equation is x = –(y + 1) + cey