Let's have 2 real continuous functions $X(z,t)$ and $Y(z,t)$ , defined for all real numbers.
It's known the equations system below :
$$\int_{\tau}^{\infty}\frac{ \partial{X(z,t)}}{\partial{z}} dt = - \frac{Y(z,\tau)}{a} $$
$$\int_{-\infty}^{\infty} Y(z,t) dz = b $$
$$\int _{0}^{\tau} \frac{\partial{Y(z,t)}}{\partial{z}} t dt = - X(z,\tau)\tau $$
, where $a$ , $b$ and $\tau$ are positive and known .
How can I find $\int_{0}^{\tau}Y(0,t)t dt $ ?
I'm new in this kind of equations. Any link with useful theory would be helpful.
Thanks.