# Calculation of evolution equation with NDSolve

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...

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}];

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])