Consider the following SDE: $dX(t)=(A(t) X(t) + a(t)) dt + (B(t)X(t) + b(t))dW(t)$

The solution is given by: $X(t)= Z(t) cdot (x + frac{a(t) -B(t)b(t)}{Z(t)}dt+ frac{b(t)}{Z(t)} dW(t))$

where $Z(t)$ is the solution of

$d Z(t) = Z(t) A(t) dt + Z(t)B(t) dW(t)$ and $X(0)=x$

To proof this, denote the right hand side of the solution by $Y(t)$, such that X(t)=Z(t)Y(t)

Applying Ito’s product rule yields:

begin{align} Z(t)Y(t)&= x+ Z(t)dY(t) + Y(t)dZ(t) + dZ(t)dY(t)\ &= x +Z(t)left((A(t) Y(t) + a(t)) dt + (B(t)Y(t) + b(t))dW(t)right) + Y(t)(Z(t) A(t) dt + Z(t)B(t) dW(t) \ &+ (B(t)Y(t) + b(t))Z(t)B(t) dW(t)

end{align}

How can I simlify thar further?