I set up this differential equation
$$ E-\tau\frac{dv}{dt} -v(t)=0$$
for an electrical network. Here $v(t)$ represent voltage. Suppose we know the beging condition (BC) for voltage ($V_0$).I reduce first the differential equation so:
$$-\tau\frac{dv}{dt} -v(t)=0.$$
The solution here is $$C\exp(-t/\tau)$$ and $C$ is the constant of integration. What I don't know is, may I fill the solution with BC at this stage to find $C$ or I should first find the particular solution, superpose both solutions and then fill the equation to find the constant?
Assume you found the general homogenous solution as
$$v_h(t)=f(t,c)$$ and adjust the constant by solving
$$f(0,c)=v_0.$$
Now the general non-homogeneous solution will be of the form
$$v_{nh}(t)=f(t,c)+g(t)$$ where $g(t)$ is a particular solution.
If you plug $t=0$,
$$v_{nh}(0)=f(0,c)+g(0)=v_0+g(0),$$ which has no reason to equal $v_0$.
[It will work if $g(0)=0$, but to achieve this you need to be lucky or adjust for the initial condition $v_0=0$, which is a little silly.]