I have near to no experience with mathematica, so this might seem pretty trivial.
I want to solve an Evolution equation, i.e. we have a function $f(t,x)$ and a differential operator $T(t,\partial_t)$. The equation is then
$T(t,\partial_t) f(t,x) = c$
As an example consider $T = t \partial_t$,the initial condition $f(1,x) = e^{x}$ and $c=1$. Then we have the differential equation
$\partial_t f(t,x) = \frac{1}{t} \rightarrow f(t,x) = f(1,x) + \log(t) = e^{x} + \log(t)$
Then I would want to plot $f$ for some fixed value of $t$, e.g. $f(2,x) = e^{x} + \log(2)$.
My first try would be something like
NDSolve[{Derivative[f[t, x], t] == 1/t, f[1, x] == Exp[x]}, {x, 0, 3}, {t, 1, 2}]
Plot[Evaluate[f[2,x] /. %], {x, 0, 3}]
But that doesn't seem to work...