On the solution of Volterra integral equation I got stuck with some strange point, solving Volterra integral equation:
$$
\int_0^t (t-s)f(s) ds =\sqrt{t}.
$$
The solution can be obtained by ssuccessive differnetiation 
$$
\int_0^t f(s)ds=\frac{1}{2\sqrt{t}}, \quad \mbox{and then}
$$
$$
f(t)=-\frac{1}{4t\sqrt{t}}
$$
But, when I substitute this solution to the original equation
$$
\int_0^t(t-s)\left[-\frac{1}{4s\sqrt{s}}\right]ds=\left[\frac{s+t}{2\sqrt{s}}\right]_0^t=\sqrt{t}-\infty
$$
I can't figure out where I was wrong. Please explain. 

I am not an expert on integral equations or the like, but the origin of the trouble is clear -- the differentiation process loses sense of a constant. The infinity comes because your function is not integrable near zero. Being only mildly acquainted with integral equations, I would try two other approaches -- Neumann series and Laplace transform -- and see if they give something reasonable and get some intuition from the result.
