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,

Let t = log x

[Differentiating both sides]

By substituting this in the above integral, we get

We have

⇒ I.F = e^{log t}

⇒ I.F = t [∵ e^{log x} = x]

∴ I.F = log x [∵ t = log x]

Hence, the solution of the differential equation is,

Let t = log x

[Differentiating both sides]

By substituting this in the above integral, we get

We know

[∵ t = log x]

Thus, the solution of the given differential equation is