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, and

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

We have

[∵ m log a = log a^m]

∴ I.F = x^–1 [∵ e^{log x} = x]

Hence, the solution of the differential equation is,

Let

By substituting this in the above integral, we get

We know

∴ y = –e^–x + cx

Thus, the solution of the given differential equation is y = –e^–x + cx