Volterra integral equation : 'literally match the independent variables' message

Question

I want to solve an Volterra type Integral equation,

and as a practice I entered the following code to my mathematica:

Clear[Func, x, t]; DSolve[
D[Func[x], x] == Integrate[Func[t], {t, 0, x}], Func[x], x]

But I got an error message,

DSolve::litarg: To avoid possible ambiguity, the arguments of the dependent variable in

(function)

should literally match the independent variables. >>

I guess that the argument of Func[...] should be always 'x' (not 't') in the above example.

I first thought changing my code to

Clear[Func, x, t]; DSolve[
D[Func[x], x] == Integrate[Func[x], {x, 0, x}], Func[x], x]

But it seems it's not the integral equation I want to solve.

Also in that case I got the following warnings,

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Are there any ways to assign variable to the bounds of the integration interval?

Thank you in advance,

Answer 1

Your ODE is $$ \left\{ \begin{array}{l} \frac{df}{dx}(x) = \int_0^x f(t) \, dt,\\ f(0) = f_0. \end{array} \right. $$

You can differentiate the first equation and obtain $\displaystyle \frac{d^2f}{dx^2}(x) = f(x)$.

The initial conditions are $f(0) = f_0$ and $f'(0) = 0$ (as follows from the first equation).

This ODE is very simple to solve and the solution is $f(x) = f_0 \cosh(x)$.

I wonder why you didn't solve the equation by hand, but tried to use Mathematica instead. You should realize that Mathematica isn't just a magic box in which you put something like

Solve["47th Hilbert's problem"]

and wait for a solution. Sometimes it's more instructive to think a bit more about the problem at hand and come up with a simple solution.

By the way, you can solve the ODE and even verify it with the following code:

sol = First@DSolve[{f''[x] == f[x], f'[0] == 0, f[0] == f0}, f, x];
f[x] /. sol // FullSimplify
f'[x] == Integrate[f[t], {t, 0, x}] /. sol // FullSimplify