Given an ordinary differential equation with degree $n$, do we include the particular solution when solving for the arbitrary constants?
For example, given the first order ODE of a circuit:
$$
\frac{\text{dv}}{\text{dt}}+\frac{v}{RC}=\frac{V_s}{RC}
$$
with initial condition $\text{v}(0)=V_0$
we have
$$
\text{v}_c=ke^{-\frac{t}{RC}}\\
\text{v}_p=V_s\\
v=ke^{-\frac{t}{RC}}+V_s
$$
The $k$ is solved by including the particular solution to yield $$k=V_0-V_s$$
The example above makes use of the particular solution. My other question is, since the ODE is linear, can't we solve for the arbitrary constants without involving the particular solution?
Best Answer
Nice question.
When you add the particular solution, that changes $v(0)$. So if you align $v(0)$ before adding the particular, $v(0)$ will no longer be correct.