How to derive the solutions of the Differential Equations when using convolution?
What you're describing is not really convolution per se, although the results can be used with convolution to get solutions for more complicated input signals. Maybe you're slightly misunderstanding what the book is trying to do?

The usual method for solving this for arbitrary \$x(t)\$ is to first solve it for \$x(t) = \delta(t)\$, which is what the calculation in your book very likely does. This solution is called the impulse response. The Fourier transform of \$\delta(t)\$ is indeed \$1\$ (or \$1/\sqrt{2 \pi}\$, depending on the conventions used), so in that sense the right hand side "is treated as one".

Then convolution is used to represent the general case as
$$
x(t) = \int_{-\infty}^\infty \mathrm{d}T\, x(T) \delta(t - T).
$$
Since the differential equation is linear and shift invariant (a sum of shifted solutions is a solution), and here \$t\$ appears only inside the delta function, you can use this to write the solution in the general case as a convolution between \$x(t)\$ and the impulse response.

In the second case, the input is the derivative of some given function \$x(t)\$, whose Fourier transform you can relate to the FT of the function itself by \$\mathcal{F}\{x'(t)\}(\omega) = i \omega \mathcal{F}\{x(t)\}(\omega)\$. I think what your book does here is that it solves the problem for the derivative of the delta function, to make a general formula for this case.