I am trying to solve the following ODE with the given boundary condition.
$${Z(t)f(t)}-\frac{d Z(t)}{d t}=0$$
with the boundary condition
$$\lim_{t\rightarrow\infty}Z(t)=0$$
I have solved in the following way:
$$\int\frac{d Z(t)}{d t}\frac{1}{Z(t)}dt=\int f(t)dt$$
$$Z(t)=e^{\int f(t)dt-C}=C_{1}e^{\int f(t)dt}$$
where $e^{-c}=C_{1}$ is constant.
How can i use the boundary condition in the solution and to get rid of the constant?
Either $Z\equiv 0$ or any solution (for any $C_1$) will solve the boundary condition. It depends upon the behavior of $\int f(t)dt$ at $+\infty$ (whether ot not it goes to $-\infty$).