Q:

Linear equation with additive noise

The following is an excerpt from Mikosch's Elementary Stochastic Calculus:

I'm trying to understand the calculation in the red box where the version of Ito's lemma mentioned is as follows:

Question:
How is (2.30) used in the red box?

I tried to apply (2.30) directly by using $f(t,x)=y(t)x$ and
$$
A_s^{(1)}=c_1(s)X_s+c_2(s),\quad A_s^{(2)}=\sigma_2(s). 
$$
But I ended up with something very different from the desired formula in the red box. (For instance, in my calculation, the term $c_1'(t)$ pops up. ) 

A:

I didn't get the desired formula due to wrong calculation of the partial derivatives. For $f(t,x)=y(t)x,$ one should have
$$
f_1(t,x)=y'(t)x=-c_1(t)y(t)x\\
f_2(t,x)=y(t)\\
f_{22}(t,x)=0
$$
A direct substitution to (2.30) would then solve the problem. 

