How can i solve the integral equation

Question

How can i solve the integral equation $$z(t) =\int_0^t z(q)(1-(t-q))\,dq+C?$$

Solving for the function $z$.

I have access to Mathematica.

Answer 1

Letting $w(T)=\int_0^T z(t)\,dt$, we see that $$w'(T)=z(T)=\int_0^T z(t)(1-(T-t))\,dt = (1-T)w(T) + \int_0^T tz(t)\,dt$$

Differentiating again, we get: $$w''(T) = (1-T)w'(T) - w(T) + Tw'(T) =w'(T)-w(T) $$

$$w''(T) = w'(T) -w(T)$$

That's a pretty standard homogenous differential equation, with two initial conditions, $w(0)=0$ an $w'(0)=z(0)$ fixed by your condition.

Essentially, this means that $$w(T)=Ce^{T/2}\sin\left(\frac{\sqrt 3 }2T\right)$$

Where $C$ is some constant determined by $w'(0)=z(0)$.