I have a set of nonlinear and coupled PDEs for a cable actuated system. I want to transfer this set to a set of ODEs and write it down in Matrix form. I used Galerkin's method and derived the ODEs in discritized form. I have a problem in converting this set of equation into a matrix form like "Mq"+Cq'+Kq=F". The derived equations are very complicated and contains some nonlinear terms. One of the three equations looks like this:

I want to use leibniz rule to differentiate the following Integral with variable domain:
A :=(t,x) -> int(Yt^2, x = 0 .. zeta);
Yt and zeta are defined as:
alias(zeta=zeta(t));
Yt:=(t,x)->diff(y(t,x),t);
When I want maple to derive D[1](A(t,x)), its answer is:
2*D[1](Yt)*Yt*Zeta+Yt^2*D[1](Zeta)
Although it uses Leibniz rule correctly, it dose not take Yt as a function of x by mistake. 2*D[1](Yt)*Yt*Zeta should had been derived as int(2*D[1](Yt)*Yt,x=0..zeta).