I am trying to understand the frequency domain and Fourier transforms by using them to solve simple differential equations. In particular, I am interested in the equation: $$ dx(t) = x(t)dt \quad \text{or} \quad \frac{dx}{dt} = -x \quad \text{with } x(0)=x_{0} $$ Clearly a solution to this is $x(t) = x_0 e^{-t}$. However, I fail to recover this result when attempting to solve using the Fourier transform.
I will use the follwing conventions for the Fourier transform: $$x(\omega) = \mathcal{F}[x(t)] = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} x(t) e^{i \omega t} dt \, ,$$ and its inverse $$x(t) = \mathcal{F}^{-1}[x(\omega)] = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} x(\omega) e^{-i \omega t} d\omega \, .$$
Fourier transforming both sides of $\frac{dx(t)}{dt} = -x(t)$ gives: $$ \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \frac{dx(t)}{dt} e^{i \omega t} dt = -x(\omega) . $$ The left hand side can be expanded using integration by parts: $$ \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \frac{dx(t)}{dt} e^{i \omega t} dt = \frac{1}{\sqrt{2 \pi}}\big[x(t)e^{i \omega t} \big]^{+\infty}_{-\infty} - i \omega x(\omega) $$ To proceed, we need boundary conditions of $x(t)$ at $t = \pm \infty$, which we do not have as part of the original problem. Normally, when dealing with Fourier transforms, we assume $x\rightarrow 0$ as $t\rightarrow \pm \infty$. If we do this, (even though this is not the correct boundary condition at $t\rightarrow -\infty$) the equation becomes: $$ -i \omega x(\omega) = x(\omega)\quad \text{ which yields } \quad x(\omega)=0 $$ Inverting the Fourier transform of $x(\omega) = 0$ yields $x(t)=0$. This is the steady state solution as $t\rightarrow \infty$, but does not yield the correct solution $x(t) = x_0 e^{-t}$.
How do I correctly solve this simple equation using the Fourier transform?