Differential integro-equation

Question

So I'm stuck with the following problem:

$b'(t)$ + $\int_{0}^{t} (t-q)b(q)dq = t$
$b(0)=0$

The book calls this an integro-differential, but I can't really understand how to solve it, currently I've put up:

b(s) + (1/s^2)b(s) = 1/s^2

I dont know where to go from here or if this is even right, any help is appreciated, thanks in advance!

Answer 1

If you set $$ u(t)=\int^t_0(t−q)b(q)dq $$ then $$ u'(t)=(t−t)b(t)+\int^t_0 1·b(q)dq=\int^t_0 b(q)dq\\ u''(t)=b(t)\\ u'''(t)=b'(t) $$ which allows you to transform the equation into an ordinary differential equation.