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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...

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You need to use D instead of Derivative, and you need to include the function to be solved for (f):

NDSolve[{D[f[t, x], t] == 1/t, f[1,x] == Exp[x]}, f, {x, 0, 3}, {t, 1, 2}];
Plot[f[2, x] /. %, {x, 0, 3}]

enter image description here

In this case, the Evaluate is not necessary.

(If you do want to use Derivative instead, it should be Derivative[1, 0][f][t, x])

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