Assume $T_j, j=1,2,3$ to be a linear operator $T_j: C^{1}[0, 1] \rightarrow C[0,1]$.
If we're given the form of the operator, i.e.,
$\begin{equation} T_1x(t) = x'(t) - x(t), \end{equation}$
it is easy to check that under the restriction of $x(0)=0$, the inverse operator $T_1^{-1}$ has the form of $\int_{0}^{t} e^{t-s}x(s)ds$.
Alternatively, we can check that for the operator $T_2x(t) = x'(t) + 4x(t)$, the inverse has the form of $T_2^{-1} = \int_{0}^{t} e^{4(s-t)}x(s)ds$ under $x(0)=0$, by following similar logic.
My question: are there any general methods/rules for arriving to the particular expressions of such inverse operators? Obviously, in cases similar to which I've mentioned one can guess the expression with some trial and error, however such approach becomes tedious when dealing with more complicated expressions of the operator $T$ (i.e., $T_3x (t) = (t^2 + 1)x'(t) + 2t^2 x(t)$).
As far as I understand, we need to solve $T^{-1}(Tx) = x$ for $T^{-1}$, however I'm not sure how to approach this problem. Any help would be appreciated!